> 







The Machinist and Tool Maker's 
INSTRUCTOR, 

CONTAINING AN EASY METHOD OF CALCULATING AND LAYING 
OUT DIFFICULT WORK IN THE MACHINE SHOP. 



THE CONSTRUCTION OF GEARING AND GEAR CUTTERS. 

THE UNIVERSAL MILLING MACHINE WITH PRACTICAL 
EXAMPLES. THE MAKING OF CUTTERS FOR VARIOUS 
PURPOSES. ALSO A METHOD of CALCULATING GEAR- 
ING FOR THE CUTTING of SPIRALS, Etc. 

THE UNIVERSAL GRINDING MACHINE 
WITH PRACTICAL EXAMPLES, 

The ELEMENTARY PRINCIPLES of MECHANICAL POWERS. 

SIMPLE AND COMPOUND GEARING. 

THE TREATMENT OF STEEL. 



TOGETHER WITH TABLES, RULES AND VALUABLE INFORMATION FOR 
MECHANICS GENERALLY. . ^ \ U9 



FULLY ,L B L T RATED V^fcK-^-\ 

EDWARD GENUNG, * * 



Mechanical Engineer and Machinist, 
NEW YORK. 



&3 



K 



COPYRIGHT, JUNK, l8 
BY 

Edward Genung. 



KJ^ 



yy.% 






Price, $5.00 






PRESS OP J. B. SAVAGH 
CLEVELAND. 



Preface. 



TN the preparation of this work I have taken into 

consideration the fact that as the average 
machinist has not had the education necessary to 
understand the more advanced mechanical works, 
and believing myself to be in a position to know the 
subjects least understood, (having had the manage- 
ment of machine shops for the past sixteen years) and 
also knowing the dislike generally for mathematical 
problems, more especially those in algebraic formulas, 
I have adopted arithmetic and plane trigonometry for 
this work, the former in as simple a manner as pos- 
sible, and in the latter treating of right angled tri- 
angles only. 

I would call the reader's attention particularly to 
that part of the work devoted to plane trigonometry, 
as, by its methods we are enabled to lay out geometric 
figures accurately and also to measure circular and 
angular distances of any description or magnitude. 
This subject, although new to most mechanics, can, 
by careful study be easily comprehended. The rela- 
tion of sines, cosines, tangents and secants in regard 
to their respective positions should be carefully 
studied. 

I trust the reader will not only recognize the 
value of this work, but that he will also appreciate its 
subjects. 

The writer gratefully acknowledges favors shown 
by Prof. De Volson Wood and R. H. Thurston; also 
to the Brown & Sharpe Manufacturing Co., and the 
Pratt & Whitney Company. 

The Author. 

New York, July, 1896. 



CHAPTER 



The following examples have been prepared to 
assist the reader not conversant with arithmetic : 

ADDITION OF INTEGERS OR WHOLE NUMBERS. 

The sign of Addition is marked thus (+), and, when 
placed between numbers means that they are to be added 
together, thus, 4 + 1 + 5 reads 4 plus 1 plus 5. 

The sign = means equal, and, 4 + 1 + 5 = 10; 6 + 2 = 8; 
1 + 1 = 2. 

The sign of subtraction is marked thus (— ), and, when 
placed between two numbers, means that the one after it is to 
be taken from the one before it, thus, 7 — 3 = 4, or 7 minus 3 
equals 4 ; 2 — 1 = 1 ; 70 — 50 = 20. 

MULTIPLICATION. 

The sign of multiplication is marked thus (X), and, when 
placed between two numbers, means that they are to be multi- 
plied together, thus, 4 X 2 = 8, or 4 times 2 are 8; 7X2X2 = 
28;2X3X2X2X2 = 48, 

INTEGERS. 



A Vinculum , or bar, and a Parenthesis ( ), both have 

the same meaning; thus, 8X6 + 3 reads 8 times 6 plus 3 ; 
now 6 + 3 are 9, and 8 X 9 are 72. 4 X (6 + 2) : 6 + 2 are 8, 
and 4 times 8 are 32, Ans. 3 X (2 + 2 + 2) = 18, Ans., 
5x1 + 1 + 1 + 1 + 1 = 25, Ans. 



6 THE MACHINIST AND TOOI. MAKEE'S INSTRUCTOR. 

DIVISION. 

The sign of division is marked thus (-=-), and, when placed 
between two numbers, means that the number on the left is to 
be divided by the number on the right, as follows : 10 -7- 2 
means that 10 is to be divided by 2 = 5 Ans., 40 -7- 20 = 2 Ans., 
74 -T- 2 = 37 Ans. 

We sometimes place them in the following manner, thus 
40/20, which means that 40 is to be divided by 20, which 
would be 2 ; 40/2 == 20 Ans., 1000/4 = 250 Ans., etc. 

DECIMALS. 

The word decimal means numbered by tens, and, in 
enumerating figures, we sometimes see a figure thus 4.6, 
which reads 4 and 6 tenths. The figure 6 to the right is called 
the decimal, or fractional number ; the figure to the left, or 4, 
is a whole number. The point between is called the decimal 
point because it separates whole numbers from parts of num- 
bers, or fractions, etc. 

ADDITION OF DECIMALS. 

What is the sum of 4 tenths, 5 hundredths, 52 thousandths? 
.4 reads 4 tenths, 
.05 " 5 hundredths, 
.052 " 52 thousandths. 

.502, Ans., reads 502 thousandths. 
Always keep the decimal point in a column, and as there 
are no whole numbers, consequently there are no units. Com- 
mencing at the left, enumerate as follows: tenths, hundredths, 
thousandths, etc. 

.001 = 1/1000, or, 1 thousandth, .021 = 21/1000, or, 21 
thousandths. 1.001 reads 1 and 1 thousandth. 

Add the following numbers : 42.1, 421, 4, .04, .044. 
42.1 reads 42 and 1 tenth, 
421. " 421 

4. " 4 

.04 " 4 hundredths, 

.044 " 44 thousandths. 



467.184 " 467 and 184 thousandths. 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



SUBTRACTION OF DECIMALS. 

From 23.1 take 19.9, or, 23.1 — 19.9. 

23.1 reads 23 and 1 tenth, 
19.9 " 19 and 9 tenths, 

3.2 Ans., reads 3 and 2 tenths. 
From 45.04 take 20.004. 

45.04 reads 45 and 4 hundredths, 
20.004 " 20 and 4 thousandths. 



25.036 Ans., reads 25 and 36 thousandths. 
Keep the decimal points in a column. 
431.0095 — 47.000095. 

431.0095 reads 431 and 95 ten thousandths, 
47.000095 " 47 and 95 millionths. 



384.009405 Ans., reads 384 and 9405 millionths. 
MULTIPLICATION OF DECIMALS. 
Multiply 3.7 by 2.6, or, 3.7 X 2.6. 

3.7 reads 3 and 7 tenths, 
2.6 " 2 and 6 tenths. 

222 
74 

9.62 Ans., reads 9 and 62 hundredths. 
840 X .004. 

840 reads 840, 
.004 l< 4 thousandths. 



3.360 reads 3 and 36 hundredths, or, 360 thousandths. 
.004 X .000004. 

.000004 reads 4 millionths, 
.004 " 4 thousandths. 



.000000016 reads 16 billionths = product. 

Always cut off as many figures in the product, or answer, 
as there are decimals in the multiplier and multiplicand 
together. There are nine in this last case. 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 



DIVISION OF DECIMALS. 

Divide 4.5 by 7.2, or, 4.5 -+- 7.2. 

7.2 ) 4.500 ( .62+ reads 62 hundredths. 
4.32 

180 
144 

36 + 

4.5 in this case is the Dividend, and reads 4 and 5 tenths. 

7.2 in the example is the Divisor, and reads 7 and 2 tenths. 
The Dividend being the smaller number we place several 
ciphers to the right, which does not change the number, but 
makes it more convenient to divide. It now reads 4 and 500 
thousandths, as 5 tenths, 50 hundredths and 500 thousandths 
are the same, or, means one-half, it will be seen that it remains 
the same. The Dividend having three decimal points and the 
Divisor but one, subtract one from three, we have two places 
or figures to cut off. 

.4 -f- 142. 

142) .40000(281 + 

284 

1160 
1136 

240 

As the Dividend in this example contains five decimal 
figures, and none in the Divisor, the answer, or quotient, must 
have five decimal places also, and as we have but three figures, 
or as shown 281, we must annex two ciphers to the left, mak- 
ing it thus, .00281, Answer. 

Remember that if we continue the quotient one or more 
figures farther, we must also, for every figure thus carried, 
place a cipher in the dividend, which would leave the answer 
the same. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



17 and 1 tenth. 



471.38 -r 


- 27.5. 








27.5 ) 471.38 ( 
275 


17.1 + 


Answer 




196 3 
192 5 






3 88 
2 75 




.0000050 


1 13 

-T-.02 








.02 ) .0000050 ( 
4 


.00025 Answer. 




10 
10 







In this example the Dividend reads, 50 ten millionths, 
and the Divisor 2 hundredths, and, as there are 7 decimals in 
the dividend and 2 in the divisor, subtracting 2 from 7, we 
have 5 decimals in the quotient, and, in a case of this kind, we 
have to add three ciphers, making the answer read 25 hundred 
thousandths. 

FRACTIONS. 

There are two kinds of fractions : Common and Decimal. 
Thus, .750 reads 750 thousandths, and is a decimal fraction, 
and if placed thus, 750/1000, it is a common fraction ; the 
meaning is not changed but still reads 750 one thousandths. 

There are also simple and compound fractions, thus, J4 of 
anything, 1/1000, 1/10, %, etc., are simple fractions. 

J4 of^£, or Yq of 1/5, is a fraction of a fraction, and is 
called a compound fraction. 

ADDITION OF FRACTIONS. 

Add the following sums : M + /^ + 1/6* 

The figures above the line are the Numerators, and, those 
below, the Denominator. We now find the smallest number 
(called the Least Common Denominator) that 4, 3 and 6 will 



10 THE MACHINIST AND TOOI, MAKER* S INSTRUCTOR. 

divide without any remainder, which is 12; now one-fourth of" 
12 is 3, one-third of 12 is 4 and one-sixth of twelve is 2, or thus, 

12 -r- 4 = 3 

12 -i- 3 = 4 

12 -r- 6 = 2 

9 
Since these are all fractional numbers, less than one, 
1/4 of 12 twelfths = 3/12 
1/3 of 12 " =4/12 
1/6 of 12 " = 2/12 
or added together the sum is 9/12 = %. 
We can also do this in decimals, thus, 
1/4 of 1000 = .250 
1/3 of 1000 = .33333 + 
1/6 of 1000 = .16666 + 

.74999 + 
Add together 2/5 + % + 1/7. 
In an example of this kind we multiply all of the lower 
figures, (Denominators,) together, thus, 5X3X7 = 105, 
then 2/5 of 105 = 42 
2/3 of 105 = 70 
1/7 of 105 = 15 

127 which means 127/105, then divid- 
ing 125 by 105 = lyVs? Answer. 

Add together the following fractions : 

.1" + 1/5" + .3" + 34 7/ > or fractions of an inch. 
1/10 of 1.000 = .100 
1/5 of 1.000 = .200 
5/10 of 1.000 = .300 
1/4 of 1.000 = .250 

.850, reads 850 thousandths of an inch. 

SUBTRACTION OF FRACTIONS. 



have 8 times 9 are 72, and % of 72 are 63 ; and 2/9 of 72 are 16. 
Then 16 from 63 = 47. As the denominators are 72, it will 
read, thus, 63/72 — 16/72 = 47/72, Answer. 



THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 11 

4J£" — 2.1", reads, from 4^ inches take 2 and one- 
tenth inches. 

41^ = 4.125 
2.1 = 2.100 



2.025, reads 2 inches and 25 thousandths. 
12%" — 2.05" ? From 12%" take 2 and 5 hundredths 
of an inch. 

12%" = 12.375" 
2.05"= 2.05" 



10.325" reads 10 inches and 325 thou- 
sandths of an inch. 

10^ -4i. 10^ = 21/2 

4| = 29/7, multiplying 
the denominators together we have 2 X 7 = 14, and we have 

21/2 of 14 = 147/14 
29/7 of 14 = 58/14 



89/14, or thus, 14 ) 89 ( 6 T % Answer. 

84 

5 

or carried out, thus, 

14 ) 89 ( 6.35+ Answer. 

84 

50 
42 

80 
70 



or in decimals, 10^ = 10.5000 
44 = 4.1428 



7 ) 1,0000 6.3572, Answer. 



,1428 
Always remember that in decimals we try to reduce to 
thousandths, thus when we say 4-J, we divide 1000 by 6 ; if 
we say 4^, we divide 1000 by 9, etc. 



12 THE MACHINIST AND TOO!L MAKER' S INSTRUCTOR. 



MULTIPLICATION OF FRACTIONS. 

}iXy 2 = Vs reads M multiplied by y 2 ; or y 2 X M = U 
multiplied by J^ = tne same. 

If we cut an apple into two equal parts, each of those 
parts is one-half, and, if we take one of those parts and cut 
it into four (4) equal parts, then there would be four (4) parts 
in the other half also, and as 4 and 4 are eight, or equal parts, 
•each one of those parts would be one-eighth (%) the answer, 
and, it will be the same as saying y 2 of J4> or > M °f Vk* ^et us 
take 34 °f % an inch for an example and see if we do not get 
y% of an inch. 

y^ of an inch = .500, reads 500 thousandths, 
M ) .500 

.125, Answer, reads 125 thousandths, or ^' 
1/2 X 1/6 = 1/12 Multiply the Numerator and Denomin- 
1/7 X 1/8 = 1/56 ator together. 
3/4 X 4/5 = 12/20 

12 X 3/8 = 36/8 = 4f or 4J£, Answer. 
12.5X .004 
12.5 
.00 4 



.050 Answer, reads 5 hundredths, or, 500 ten thousand- 
ths. 

In multiplying decimals the most important thing to re- 
member is where to place the decimal point. 12.5 in the ex- 
ample is the Multiplicand. 12 is a whole number and .5 is 
the decimal; consequently, a decimal point must be in front 
•of it. .004 = 4/1000 is also a decimal and has three figures, to- 
gether with the one above, making 4 figures to cut off in the 
answer. Should there not be as many figures in the answer to 
cut off as in the Multiplier and Multiplicand together, then 
always place as many (Prefix) as required to do so and place 
Ihe decimal point in front as shown below. 

5/12 X 7/9 = 35/108 Answer. 

27 X y Q = 27/3 » 9 Answer. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 13 



I .004 X .004 reads 4 thousandths multiplied by 4 thousand- 
ths. .004 
.004 



.000016 Answer. 

.000002 X .006. .000002 

.006 



.000000012 Answer. 

DIVISION OF FRACTIONS. 

1 -r- 34 = 4 Answer. 

34 -5- 1/5 reads 34 divided by 1/5. 

34 -r- 1/5 = 5/4 = 134, Answer. 

10f -5- J4 lOf = 74 A. Now if we had said lOf -r- 3, 
we would take J£ of the 74/7 or 74/21 ; but we say divided 
by 3^3, which would be 9 times as much, since y% is only 
1/9 of 3, so we place the figures thus, 74/7 -f- 3/1, or invert 
the Yq ; now multiply the numerators together and divide by 
the denominators, thus, 74 

3 

7) 222 

31f Answer. 
If I had said divide 12" X Yz\ or divide 12 inches by 3^ 
inches, you would not say the answer was 6, but you would 
say 24 ; the answer is, however, the same, no matter what we 
are speaking of ; again 
48-t-4 = 12 Answer. 

48 -r- 34« As there are four fourths to one, in four whole 
numbers there would be four times four or sixteen times as 
much ; now let us invert (turn upside down) the figures, and 
see what we get. It should be 16 times 12 or 192. 
48 -H 4/1 = 192/1 or 192, Answer. 
8-rl/3 = 24 Answer. 
1 -T- 1/10 = 10. 
4 -f- 1/10 = 40. 
7 -T- 1/7 = 49. 
43^-M = 18. 
2^-^ = 5. 



14 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 

INVOLUTION, 
or, Squares, Cubes, etc. 

The word Involution means the raising of a quantity to 
any given power. 

The power of any number is obtained by using that num- 
ber one or more times as a factor, thus, 2 X 2 = 4, means 
that 4 is the square of 2, or the second power of 2 ; the first 
power of the number is 2 or the number itself. Again, 
4 X 4 X 4 = 64. 

4 =, the first power, 

4 X 4 = 16, the second power. 

4 X 4 X 4 = 64, the third power. 

The second power would then be the square of the num- 
ber, and the third power would be the cube of the number, etc. 

The squares, cubes, etc., are also indicated in another 
manner, thus, 4 2 means the square of 4, or 4 X 4 = 16, Ans. 

4 3 means the cube of 4, or 4 X 4 X 4 = 64 Ans. 

2 5 means the fifth power of 2, or 2 X 2 X 2 X 2 X 2 = 32 

5 4 means the fourth power, or 5x5x5x5 = 625 Ans. 

The small figure at the top of the number is called the 
Index, and means the power. 

Remember the figure itself is always the fii st power. 

EVOLUTION. 

Square and Cube Root. 

The word Evolution means the reverse of Involution. 

The sign of square root is the symbol y/ and is always 
placed before the number, thus, \/3, which means that the 
square root of 3 is wanted; they are usually made thus, 2 \Zo> 
to distinguish them from cube root, which is also made in the 
following manner z y/2>. The small figure 2 means the square 
root and the small figure 3 means the cube root is wanted of 3. 

What is the square root of 4, or 2 \/4? I explained in 
Involution that the square of 2 was 4, and, Evolution being 
the reverse of Involution, the square root of any number is 



THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 15 

found by the following method. In the example the figure 2 
multiplied by itself, or 2 X 2 = 4, now 2 is the square root cf 
1, and 3 is the square root of 9 : 

2^/16 = 4, means the square root of 16 is 4, 

2 ^/64 = 8, means the square root of 64 is 8. 

The square root of 9/25 = 3/5; of 16/64 = 4/8 ; of T 2 D % = 
b/10 ; of 4/9 = 2/3 ; of 4/16 = 2/4, etc. 

Find the square root of 256. 
2 56 ( 16 Ans. 
1 

156 
26 156 
In square root we commence by separating into periods ci 
two figures each; always remember to commence at the 
decimal point to cut of, thus, in 256 there is no decimal point, 
because it is a whole number, then commence from the right. 
Extract the cube root of 25.4. 

25.40 (5.039+ Answer. 

25 

4000 
1003 3009 



99100 
10069 90621 



8479 
Find the square root of 18. 

IS'00 (4.2426 + 

16 



82 ) 


2 00 
1 64 


S44) 


3600 
3376 



8482 ) 22400 
16964 

81846 ) 543600 
509076 



16 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

Find the greatest square that will go into 18, which is 4 ; 
5 would be too much because the square of 5 is 25. Place the 
4 in the quotient, as in division ; subtract 16 from 18, we have 
2, and annex the next period, or add two ciphers in the 
absence of other figures. Now double the figure 4 and 
place it on the left, as shown, to form a new divisor. Now 
18.00 ( 4 find how many times this 8, with sor*.e other 
16 figure will go into 200, which is 2; place this 

2 at the right of 8, making it 82, then multiply 

8 ) 2 00 the 82 by the 2, which is supposed to go into 

the quotient with the 4, and continue as before. Remember 
to double all figures in the quotient as fast as you proceed, 
except, the last named figure. 



CUBE ROOT. 

As square root is separated into periods of two figures 
each, so is cube root separated into periods of three figures 
each. As the cube of 2 is 8, so the cube root of 8 is 2. The 
cube of 3 is 27, so the cube root of 27 is 3, and the cube 
root of 27/64 = %, etc. 

The cube of 1 = 1, the cube root of 1 = 1, 

The cube of 2 = 8, the cube root of 2 = 1 .259 + , 

The cube of 8 = 512, the cube root of 8 = 2. 

The cube root of 512 = 8. 

The cube root of 8 = 2. 

The cube root of 1 = 1. 

The cube of 1.259+ =2. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



17 



Extract the cube root of 1420. 



1'420 

1 


(11.23 + 


12 x 300 = 300 

1X30X1= 30 

1 2 = 1 

331 


420 
331 




112 x soo = 36300 
11 X 30 X 2 = 660 

2 2 = 4 
36964 


89000 
73928 





112 2 X 300 = 3763200 

112 X 30 X 3 = 10080 

3 2 = 9 



15072000 



11319867 



3773289 

3752133 
Fstract the cube root of 9840. 

9.840 ( 21.43 nearly. 



2 2 X 300 = 1200 

2 X 30 X 1 = 60 

1 2 = 1 



1840 



1261 

21 2 X 300 = 132300 

21 X 30 X 4 = 2520 

4 2 = 16 



1 261 
579000 



134836 



39344 



214 2 X 300 = 13738800 

214 X 30 X 3 = 19260 

3 2 = 9 



13758069 



39656000 



41274207 



TO EXTRACT THE CUBE ROOT. 
Commence at the Decimal point, if any; if there is none 
ommence from the right, and separate the figures into periods 
f three figures each, as shown in the examples ; find the 
lighest number that, when cubed, will go into the first period 



IS THE MACHINIST AND TOOT. MAKER'S INSTRUCTOR. 

at the left, which in the last example is 2; place this figure in 
the quotient, then cube it, which is 8, and place it under the 
figure 9 as shown ; subtract this, and we have 1840 for a re- 
mainder. We will now have to find a divisor for this 1840, 
which is done in the following manner : Place the same 
figure in the quotient, whatever it may be, to the left of the 
example, square it and multiply by 300; then take the same 
figure and multiply by 30, find how many times these 
figures (that is, the 2 2 X 300 + 2 X 30) will go into the divi- 
dend (1840 in this instance) and we find it to be 1; then place 
this figure 1, which is called the last figure of the root, in the 
divisor, which now reads 2 X 30 X 1 ; take the same 
figure 1 and square it as shown, add all together and we 
have 1261. Then as the last figure in the. quotient is 1, we 
multiply 1261 by 1 and place it in the dividend under the 1840, 
then subtracting, we have 579 ; add the next period, or, 
if none, three ciphers, and we now proceed to find a new 
divisor, which is done in the same manner as before. 

Remember when making a new divisor that we always 
take all the figures in the quotient, no matter how many there 
are and first square them, then multiply by 300 ; next take the 
same figures of the quotient and multiply by 30, and this 
again by the next figure that we will have to find the same as 
Defore explained, and proceed as shown in the example as far 
as desired. 

MENSURATION. 

The word mensuration means the process of taking 
dimensions, and includes lines, angles, surfaces and solids. 
Surface measurements have no thickness. 

Example : We want to find how many yards of carpet to 
cover a floor that measures twenty by twelve feet (20' X 12'): 
twenty times twelve is 240 ; this would be square feet, and as 
a square yard is three feet each way, which makes 9 square 
feet, dividing the 240 by 9 we have 26f or 26% yards. 



THE MACHINIST AND TOOIi MAKEB's INSTRUCTOR. 19 

We have a table, the top of which measures two feet each 

7Q.y> and two times two are four, which makes four square 

Ifeet. This can easily be proved by running a line across the 

Imiddle on each side and it will just divide it, leaving four 

[equal spaces, which will measure a foot each way. 

To find the length of one side of a right angled triangle, 
[when the length of the other two sides are given. 

Example : Figure 1 represents a right angled triangle, 
so called because there is one right angle, or 90°. In the 
figure one of the sides is marked 2" long, the other 3 /7 ; either 
one of these sides can be called the base and the other would 
then be the perpendicular. A perpendicular line always 
means a line at right angles to the object spoken of. If we 
drive a nail in the side of a building, we call it perpendicular 
to the wall. The hypothenuse of a right angled triangle is 
always the side opposite the right angle and is always the 
longest side. To find the length of the hypothenuse we pro- 
ceed as follows : We square the base and perpendicular, 
separately, and add them together, then extract the square 
root, thus, 

2 2 or 2 times 2 are 4 
3 2 or 3 times 3 are 9 

13'00 (3".605 + Answer. 



66 ) 4 00 
3 96 

72 ) 40000 
36025 



7205 ' 



Figure 2 is similar to figure 1. In this case we have 
decimals to deal with, otherwise it is the same. 



20 THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. 






' ** / A 
/ / 

/ / 


/ / 

/ / , 
/ / ' 


,0° 


/ 1 





a 



■+--,756 — > 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 21 



Example, Figure 2 : Let us call the bottom of the figure 
the base, then If" = 1.375" : 1.375 

1.375 



6 875 
96 25 
IJ" = 1.875, which squared 412 5 

1375 

equals 3.515625, and both added 1.890 625 

3.515 625 



together == 5.406 250 ( 2.325" 4- Ans. 
4 



43 ) 1 40 
129 



462 ) 1162 
924 



4645 ) 23850 
23225 



In Figure 3 we have given the base as .756 // in length, 
the hypothenuse as 1.190 in length. In examples of this kind, 
figures 3 and 4, we square both numbers as before, but sub- 
tract the less from the greater and then extract the square 
root; the quotient is then the answer. Thus, figure 3 = 
.756 1.190 1.416100 

.756 1.190 .571536 

Ans. 



4536 


10.710 


.844564 ( .919 + 


3780 


1190 


81 


5292 


119 


181 ) 345 
181 


.571536 


1.41610 


1829 ) 16464 






16461 



Figure 4 is done in the same manner as figure 3. 

To find how long a piece of wire it will take to make a 
spiral spring, or, what is the same thing, to find the length of 
a spiral groove, sometimes called a helix. 



22 THE MACHINIST AND TOOIL MAKFe'S INSTKUCTOE. 

Example : A spring 2 //f in diameter, 1/4" pitch = 4 to 
the inch, 12" long. Diameter 2", the circumference would be: 



3.1416 

2 


1/4 


" pitch = .250" 
.250 


6.2832 squared 
6.2832 


12500 
. 500 


12.5664 
188 496 
5026 56 
12566 4 
376992 


.062500 
39.478602 

39.541102 (6.288 + 
36 


39.4786 0224 


122 ) 3 54 
2 44 




1248 )1 1011 
9984 


th of one coil 


12568 ) 102702 
100544 



= 6.288" 

48 coils 



50 304 
25152 

12)301.824 // 

25.152 Ans., in feet and decimal of a foot. 

To understand how to figure the contents of solid bodies 
is of the greatest importance to any mechanic. Frequently we 
wish to get some particular size and shape forged in the 
blacksmith shop and not having anything on hand of that 
particular shape, we have to take something smaller, larger, 
or, the shape may be quite different. It sometimes happens 
that we want a square block forged and we have to take a 
piece of round material to make it. The question then arises 
how long a piece will have to be cut off, so that we may get 



THE MACHINIST AND TOCXL MAKEE'S INSTRUCTOR. 23 



what we want without waste of material, or without having it 



&9 5 




too large, etc. Figure 5 
is a block, we will say of 
steel, as shown, 5 // square 
and 3" thick, and, as 
shown before, the sur- 
face measure would be 
thus, 



25 7/ this would be the 
3 number of square 
"^7" inches, on the end 
only, and if it is 
three inches deep or thick, we multiply this by three inches^") 
giving 75. This is the number of cubic inches in the block. 
Or, the rule would be, to multiply the three dimensions to- 
gether ; if the figures given were inches, the answer would be 
inches, and if they were feet, the answer would be feet, etc. 
If the answer obtained were feet, multiplying by 1728 would 
give the number of cubic inches. Figure 5 therefore has 75 

cubic inches. 

Now Figure 7 is a round piece of 4" 

u 4* ? *BifL *1 diameter ; let us see how many inches 
of this it would take to make figure 5. 
In any round piece we square the diam- 
eter and multiply by .7854; this gives the 
surface of one end only, thus, 




4 2 = 4 
4 

16 



.7854 
16 

47124 

7854 

12.5664 



Always remember to cut off as many figures in the answer 
as there are decimals, which is four in this case. Now there 



24 THE MACHINIST AND TOOX. MAKER'S INSTRUCTOR. 

are a little more than twelve and a half inches (12J£"), and 
dividing the seventy- five (75) by the twelve and a half, will 
give the length of a round piece of 4 // diameter, as shown in 
Figure 7. 

12.56) 75.000 (5.97" + Ans. 

62.80 



12 200 
11304 



8960 
8792 



Now if we were to get this forged to finish we would have 
to change our sizes or we would not get what we wanted; in- 
stead of calling the block that w T e intended to finish 5 /7 , we 
would say 5.2 7/ square, or more, and 3}^ // , or more, in length, 
depending upon the accuracy of the forging. There is also a 
little waste in the forging of anything in the fire, but that 
does not usually amount to much on small pieces. 



Figure 


6 is 1%; 


" X m" 


and 


w 


long; what 


is the 


contents 


in 


cubic 


inches ? 


3.5 
2.5 

17 5 

70 










8.75 
5.5 


square inches, 
long. 




4375 
4375 









5^6 




48.125 cubic inches. 

Figure 8 represents a round ball, say 4%" in diameter 
What is the number of cubic inches in the ball ? 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 25 



Cube the diameter and multiply by .5236 : 



4.25 
4.25 



s& s 




2125 
85 
1700 

18.06 25 
4.25 

903125 
361250 
722500 

76.765625 
.5236 

460593750 
230296875 
153531250 
383828125 

40.1944812500 Answer. 



Figure 9 represents a ring of 
round steel or iron, say 12" inside 
diameter and made of 2J# '' iron ; 
what is the number of cubic inches 
in the piece ? To the inside diameter 
of the ring, which is 12 // , add the 
thickness of metal 2^ // which makes 
143^ // , and this sum multiplied by 
3.1416 : 



m? 




26 



THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 



3.1416 
14.5 










2.5 Diameter of 
2.5 


ring 


157080 
125664 
31416 


the 


length 


of 


ring 


125 
50 

6.25 

.7854 
6.25 

39270 
15708 
47124 




45.55320 = 
4.908 + 

36442560 

40997880 
18221280 




9r>Q K7R1ftRfiA i 






4.908750 sq. inches 
at end of ring; if cut open and 
this multiplied by the length 
of ring, it will give the answer 
in Cubic inches. 



Figure 10. This is an eliptical ring, 8" 
between centers, 4 7/ inside diameter, and 
8 2 r/ iron; what is the cubic contents of link ? 
4 + 2 = 6", the average diameter of link^ 
**- which multiplied by 3.1416 will give the 
length of the two ends if placed together 
with the center (8 /7 ) taken out. 



3.1416 



18.8496 Now add the two center pieces of 8" or 16 / 
16 



34.8496 = the whole length of piece. 

The area of the ring is 2 X 2 X .7851 = 3.1416 and this 
again multiplied by the length, thus, 34.849+ X 3.1416 = 
109.427 cubic inches in the ring. 






THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



27 



Figure 11 represents a cone 6 /7 diameter at trie base and 
14" high. What is the number of cubic inches in the piece ? 

6" Diameter of base. The height is 14" 

6 .7854 3 ) 14 

— 36 
36 



47124 
23562 

28.2744= area of base. 



4.666 
28.27 

32662 
9332 

37328 

9332 




%// 



¥_ 



131.90882 Ans. 
Multiply the area of the base by 
one-third (J£) the perpendicular 
height. 

The following two examples are similar (in shape) al- 
though not round, to figure 9. Suppose we want to make a 
ring gear that is to be shrunk on a cast iron center, or flange. 
The ring is to finish 241^ inches outside diameter and 22% 
inches inside diameter and \\^ inches wide. In a ring as large 
as this there should not be less than J£ of an inch all over to 
finish. The best way to do a piece of work of this kind is 
to cut from a bar a piece that, when split through the center 
and opened up round, there will be no weld, leaving the ring 
solid throughout. The question now is, how ~arge a piece 
shall we use to make this ring? 

If we make it y%" large all over, then the ring forged 
would be 24^ /7 diameter outside and 22%" diameter inside 
and 1J£" wide ; then 22% from 24J£ leaves 1%", and one-half 
of this will be the right thickness of the ring when forged. 
But if we can get a bar 1%" X lj^", it will be just right when 
opened up, for there is very little waste in this kind of work. 
The ring when forged will be about |f" X 1J£" (in section). 
We will now have to find the solid contents or cubic inches in 
this ring before we cut off the bar, for we might make a mis- 
take and cut it off an inch or two too long or too short. We 
first find the diameter of the ring midway between the inside 



28 THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. 

and outside diameters, which would be about 23H'' or 23.812 
inches diameter, and multiplied by 3.1416 = 74.8 + inches 
long. As a section of the ring be ff" by 1J^ 7/ , we multiply 
this together, thus, j-f = .938 and 1%= 1.5, then .938 X 1.5= 
1.4 + square inches of section in the ring. We then multiply 
the length by the area or section, thus, 74.8 X 1.4 = 104.7 
cubic inches in the forged ring. Now, as the rough bar is 
1% 7/ X lj^ inches, we will have to find the area (or cross sec- 
tion), thus, 1%= 1.875 and 1^ = 1.5, then 1.875 X 1.5 = 2.8 + 
inches area. We now divide the cubic contents of the forged 
ring by the area of the rough bar, thus, 104.7 -5- 2.8 = 37.2 + 
inches in length, or, say 37^ // that we would have to cut from 
a bar 1% X 13^ to make the ring. 

We want to make a steel ring that will finish 16^2 // out- 
side diameter and 14% /r inside diameter and 1J4" wide. What 
size bar shall we use and what should be its length ? 

Allowing y%" all over to finish, then the forged ring will 
be 16«% X 14 J£ inches, and, measured as before through the 
center of its section, will be about 15% /7 diameter; then 15.875 
X 3.1416 = 49.87 + inches for the length of ring. As the 
forged ring should be lj^" wide, the cross section of this ring 
would be one-half (J£) the difference between the inside and 
outside diameter, multiplied by 1%, or thus, 1.12 -f X 1.5 = 
1.68 + square inches of section. Then, 49.87 X 1.68 = 83.7 + 
cubic inches in the ring when forged. Our bar should then 
be the difference between the outside and the inside diameter, 
or 2.25 inches by ]J£"; then the area will be 2.25 X 1.5 = 
3.37 + inches. Then 83.7 -r- 3.37 = 24.8" Answer. 

TRIGONOMETRY. 

Trigonometry is the science of determining the sides and 
angles of triangles by means of certain parts which are given. 

It is of the greatest importance that every machinist, and 
more especially those who work to close measurements, should 
fully understand this science. 

In this work I propose to treat of right angled triangles 
only, as I believe this will be sufficient for all purposes. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 29 

An angle is the opening between two lines that meet each 
other. The point at which they meet is called the Vertex; 
thus in Figure 12, A is the vertex of the angle. 

In Figure 21, when speaking of the angle A, or whose ver- 
tex is at A, remember to always say the angle B A C or C 
A B ; in other words, the letter that is placed between the 
other two is the angle to be considered. If at any time we 
see on a drawing an angle similar to Figure 21, with the words 
written, the angle A C B, or B C A, we would know with- 
out asking any questions that the angle was 90 degrees, as 
shown. 

The size of an angle does not depend on any length, but 
the rapidity with which the lines separate from each other. 

Adjacent angles are so called from their having a common 
vertex ; thus, in Figure 17 B is the vertex of both angles and 
B D a side of both angles ; consequently either angle is ad- 
jacent to the other, that is, the angle C B D is adjacent to 
the angle DBA. 

Angles are measured by making their vertex the center of 
a circle, and computing the arc that is included between its 
sides. 

The circumference of any circle is divided into 360 equal 
parts, called degrees, each degree into 60 equal parts, called 
minutes, aud each minute into 60 equal parts, called seconds. 

TANGENTS. 

A tangent is a straight line that touches a curve. In connec- 
tion with an angle, the tangent of an angle is a straight line 
that touches the circumference of a circle at one end, and ex- 
tending out to and meeting a secant at the other. It is always 
at right angles (or perpendicular) to the radius. 

A secant is a straight line that is drawn from the center 
of a circle and, proceeding through the circumference, meets 
with a tangent to the same circle. 

A sine is a straight line drawn from the end of an arc and 



SO THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



57* /7 



Ji-QlS 





THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 31 

touching a line perpendicular to the diameter at the other 
end. 

The cosine of an arc or angle is a line extending from the 
sine to the center. 

The versed sine of an arc or angle is a continuation of 
that part of the radius from the cosine to the circumference. 

Figure 18 will more fully explain the positions of the tan- 
gents, sines, cosines, secants, etc., with reference to the arc 
and angle, and always refers to the angle at the center of the 
circle, or as shown at C, which in this case is 50 degrees. In 
any triangle, right angled or otherwise, there are always two 
(2) right angles; consequently in Figure 18 the angle A C B 
being 50 degrees, then the other acute angle (there are always 
two in every right angled triangle), whose vertex is at B, is 
the difference between 50 and 90 degrees, which is 40 degrees, 
and this is always called the complement of the angle; that is, 
in speaking of any right angled triangle, suppose we find one 
of the acute angles to be 20 degrees, then the complement of 
that angle would be 70 degrees. 

A Chord is a straight line joining the extremities of the arc 
of a circle, and when passing through the center is equal to 
the diameter of that circle. 

The Chord of an angle is found by doubling the sine of 
half that angle; that is, if we want to know the chord of 50° 
we find the sine of 25° and double it. 

A tangent is always outside of the arc or circle, as shown 
in Figure 18, and the sine is always within the circle. 

In Figure 19 1 have shown a right angled triangle, in 
which the radius of the circle corresponds to one of the sides 
of the angle. 

Now, let it be remembered that when the distance C B 
(which is the longest line) of any right angled triangle is 
known, then the line A B is always the sine, and A O is 
always the cosine of the angle. 

Remember, also, that if the arc was removed that either 
C or B may be used as the vertex of the angle. In the 
figure (19) C is the vertex of the angle AC B, then the lines 



32 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




A S 







"\ 


*s° , 




\ 


/,'■ 








THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. do 

marked sine and cosine are correct, but if we were to figure 
the complement of this angle, then the vertex of the angle 
would be at B, the sine, and cosine would then exchange 
places, because the sine never touches the vertex of the angle 
while the cosine always does. 

Remember that when the longest line of the angle is 
known that it is then called radiu9, and that the other two 
sides are always sine and cosine. 

In Figure 20, which is similar to Figure 19 (as regards the 
angle), when the distance A C is known, then A B becomes 
tangent, and B C the secant of the angle. 

Remember that when the longest line is not known it is 
always the secant of the angle. 

Remember, also, that if the arc of the circle was removed 
that either A B or A C can be called radius (that is, if the 
distance of either one of them was known). The length or 
distance of the one known is called radius ; thus, when the 
distance A C is known it is called radius, and A. B. becomes 
the tangent of the angle, but if A. B. was known it would be 
called radius, and A C the tangent of the angle ABC. 

In other words, where we figure tangents, the longest line 
of the angle is always the secant. 

And when we figure sines, the longest line of the angle is 
always the radius. 

Also that radius and tangent are perpendicular, or at right 
angles to each other. 

And that the sine and cosine of an angle are always at 
right angles to each other. 

In the trigonometric tables that will follow this chapter 
we take the radius of 1 and, for convc ience, we usually say 
one (1) inch. 

In Figure 21 the radius of the angle is marked 3 inches, 
and the angle is 33° 40'. In the table of tangents we find that 
the angle 33° 40 / corresponds with the decimal .666 + , which 
means that for every inch in length of the radius, that the tan- 
gent is .666+ inches in length; therefore, multiplying this 
decimal by 3, we have 1.998+ for the length of the tangent. 



34 THE MACHINIST AND TOOL MAKER'S INSTEUCTOE. 



"WnillBeT 



:>6 



.- A'— > 



\B f 



<pjf§2> | 



<^/> 27 



THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 35 

In the table of secants we find the number 1.201 + oppo- 
site 33° 40', which means that for every inch in length of the 
radius that the secant is 1.201+ inches; therefore, multiply- 
ing this number by 3, we have 3.603+ inches for the length 
ofAB. 

As the angle C A B is 33° 40', the complement of this 
angle must be 90° — 33° 40' = 56° 20'. 

Figures 22, 24 and 25 are respectively 60°, 20° and 45°. 
Find the lengths of their tangents and secants. You will find 
in the table of tangents for 45° that 1 is the number given, 
which always means that the tangent is equal in length to 
radius. 

In Figures 26 and 27 we have two pieces of taper work, 
which is a very common thing in the machine shop. Figure 
26 represents a piece 10 inches long (through its axis), and 
tapers from 1%" to l%" t which is just % 7/ taper in its whole 
length. Then, dividing %" or .500" by 10 inches, we have 
.050" for one inch long ; but we only want one side of the 
taper and, consequently the taper for one inch would be 
.025". In the table of tangents we find the angle 1° 26' to 
correspond with this number (.025), and this would be the 
angle to turn or grind the piece. 

In Figure 27 the taper is from lj^" to 2J£ // , or one inch 
to 8 inches in length; this would equal .125 to each inch, or 
.0625 inches for one side, and the tangent .0625 corresponds 
to the angle 3° 35 r , which is the correct angle to set over the 
table on the Universal Grinding machine, or the taper attach- 
ment on the lathe, as the case may be. 

It must not be forgotten that if told to turn or grind a 
piece of work 2° taper, that in the machine you would set it 
for 1° only; the taper of Figure 27 is, therefore, 7° 10'. 

Now, if you were given a piece of work, say like figure 27, 
to turn 8 inches long and lJo // diameter at the small end, the 
other end not being given, you would naturally want to know, 
the first thing, how large the piece would be on the other end. 
As the angle given would be 7° 10', take % of this, or 3° 35'. 
In the table of tangents opposite 3° 35' you will find the 



36 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 37 

decimal .0626, which, multiplied by the length, 8", = .5008, 
then double this and it will tell you how much larger the 
piece must be on the other end. 

Figure 28 is also a piece of taper work quite common in 
the machine shop. In this case the taper does not extend the 
whole length of the piece. 

In turning with a taper attachment or grinding by swivel- 
ing the table of a Universal Grinding machine, the angle of 
the piece is all that is required ; that is, if the piece is longer 
or shorter, it makes no difference in the angle. But if turned 
in the lathe by setting the tail stock over, then, if a number 
of pieces are turned taper, they will all have to be the same 
length, otherwise their tapers will vary. The tail stock should 
not be set over (out of alignment) if possible, as it is hard to 
get a piece of work accurate in this manner, for the reason 
that the center has but little bearing surface. If such is re- 
quired, however, Figure 28 will show a very good way to do it 
if accuracy is desired. 

The drawing shows that the taper is to be y^" to 3% 7/ in 
length, or %" for one side; then, dividing J4" or «250 by 3.75, 
we have the decimal number .0666 + for the taper to one inch 
in length, and multiplying this by the whole length of the 
piece, which is 9 inches, we have ,5994 // , which is the distance 
that the tail stock center will have to be moved over. Now 
file a piece of wire to the micrometer caliper of this length, 
and, placing a tool with a smooth end in the tool post, with 
this piece between it and the center, then slack the screws and 
first take out the piece of wire (.odd"), then move the center over 
until it just touches the tool, then clamp the center firmly, and 
you will have but little trouble in fitting the piece by filing. 

POLYGONS. 

We now come to a class of work that is almost an every- 
day occurrence, at least in some shops. Suppose that we want 
to turn a piece of work that will be large enough to mill a 
hexigon S}^ // across the flat sides. In Figure 29 the distance 
AB is, of course, 1% inches, and is the radius of the angle 
A B C of 30°; then B C is the secant, and in the table of 
secants we find that the secant of 30° is 1.154+ , which, multi- 
plied by the radius and doubled, or by the diameter, thus, 
1.154 X 3.5 = 4.039 inches, the diameter across the corners. In 



38 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 





<- MZ 




> 


u 






^r 


V 








N 








V 








\ 








s 








T" 








' \ 








/. ^ 


o 






/ ° / s 


CO 




.,..-<- 


1 

„L..* :.>cq 


$ 














- * • 








o S 






» 


° / 






\ 


cr 






\ 


\ '' 






\ 


^ 






\ 


/ 




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\ 


/ i 




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THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



39 



other words, to find the diameter across the corners of any 
size of hexagon (six sides), multiply the secant of 30° by the 
diameter across the short side (or flat). 

Or, if you have a round piece of steel and want to know 
how large a hexagon you can mill on it, multiply the cosine 
of 30° by the diameter of the piece, and the result is the width 
across the flat side. 

Figure 30 represents a piece 2J^g inches square. 

To find the diameter across the corners of any given 
square, multiply the secant of 45° by the diameter required 
across the flat side. Thus, in Figure 30 the square required 
is 2V& inches, and the secant of 45° is 1.414, which, multiplied 
by 2^, = 3.003, the answer across corners. 

To find how large a square we can make from any piece of 
round material: Multiply the cosine of 45° (.707+) by the 
diameter of the piece given, and the result is the width across 
the flat side, or the size of square. 

Figure 31 represents an octagon or eight sided figure (or 
polygon). ^ 

To find the diameter across • *****.. 
the corners of any size octagno: 
Multiply the secant of 22^ 
degrees = 1.082+ by the 
diameter required across 
the flat side. Thus, 
suppose that Figure 
31 was to be 3 ins. 
across the flat side, 
then 1.082 X 3" 
= 3.246 inches 
across corners 
etc.; or, sup- 
pose that we 
have a round 
piece of stock 
and we want to 
mill it to the 
shape of an octa- 
gon. Then mul- 
tiply the cosine of 
22^° = .9239,bythe 
diameter of the stock, 
and the result is the diam- 
eter across the flat (some- 
times called the short diameter. Thus, if we have a piece of 
round stock 2 /7 in diameter, .9239 X 2 = 1.847+ for the size of 
octagon that could be made from a round piece 2 ins. in diam. 




40 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




Figure 32 is also a polygon of ten sides, generally called 

a decagon. 

Example : We want to mill a piece 2J^ // across the flat 

side (the short diameter). 
How large a piece of round 
stock will be required to 
make it ? Multiply the se- 
cant of 18° = 1.051+ by the 
flat diameter required; thus? 
if we want it to measure 
fyz' across the flat sides, 
1.051 X 2.5 = 2.628, for the 
diameter across the corners, 
etc. 

To find the diameter 
across the flat side of any 
decagon that many be re- 
quired, multiply the cosine 
of 18° = .951 by the diameter 

of the round piece; thus, a piece of round stock two inches in 

diameter will make a decagon .951 X 2 = 1.902 inches for the 

short diameter. 

Figure 33 is a polygon of 
twelve sides (called a dodec- 
agon. Having the short 
diameter (across the flat 
sides), to find the diameter 
across the corners multiply 
the secant of 15° = 1.035 by 
the diameter across the flat 
side. 

To find how large a dodec- 
agon we can make from a 
round piece of any given 
size, multiply the cosine of 
15° == .966 by the diameter 
of the round piece required. 




EHE MACHINIST AND TOOl, MAKER'S INSTRUCTOR. 41 




42 THE MACHINIST AND TOOL MAKER 5 S INSTRUCTOR. 



SCREW THREADS. 

There are three forms of screw threads now generally used 
in making taps, dies, machine screws, etc., and it is very im- 
portant at times to know the exact depth of the thread. 

In Figure 34, I have shown the common sharp V thread, 
in which the surfaces are inclined toward each other at an 
angle of 60°. A section of this thread, ABC, forms an 
equilateral triangle, each side of it being equal to the pitch of 
the thread (that is, in the figure the pitch A D is of the same 
length as A B or B C), and its depth, measured perpen- 
dicular to the axis, is found by multiplying the cosine of the 
angle of 30° = .866 by the pitch of the thread ; thus, in a tap 
of eight threads per inch the pitch of the thread (or distance 
between two consecutive threads) is }/% inch, or .125", then 
.866 X .125 = . 10825" ; this, then, is the depth on one side 
only, so that in any thread of J^" pitch, or eight threads per 
inch, the depth of both sides together would be twice this, or 
.2165". 

The most convenient way, however, to find the depth of 
both sides is to double the cosine .866, and we then have the 
constant number 1.732, and this number divided by the num- 
ber of threads per inch of any V shape threads, will be the 
answer ; thus, 1.732 -?- 10 threads per inch = .173" for ten 
threads ; 1.732 -4- 40 = .043 for 40 threads. 

Suppose we have a screw 6 inches diameter and we want 
a thread of 1 inch pitch cut on it (sharp V style), what will be 
the diameter of the screw at the bottom of thread ? 1.732 -r- 1 
= 1.732; then 6" — 1.732 = 4.268", Answer. 

If we have a tap one inch in diameter and ten threads per 
inch, then 1.732 -*- 10 = .1732 ; and 1" — .1732 = .826, the an- 
swer (or diameter at root of thread). 

Figure 35 is the shape of the United States standard, or 
the Franklin Institute thread (sometimes called the Seller's 
system), and the depths of threads are found in the same man- 
ner as the sharp V, and then subtracting J4 of the whole 



:he machinist and tool maker's instructor. 



43 



I 

I 



CO 

* 






I 
I 

I 
I 



"K 



44 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

depth ; thus, in the sharp V thread, four threads per inch, the 
depth on both sides is .433 // , and for the U. S. standard it will 
be .433" — X = .325", Answer. 

Figure 36 represents the Whitworth or English Standard 
shape of thread, the angle of which is 55°, as shown. We can- 
not find the depth of C B in this shape of thread in the same 
manner as shown in the previous examples, for the reason 
that the distance, A B, is not the same as A D and, conse- 
quently, we have only the angle and distance C A (which, of 
course, is one-half the pitch) to work from ; we therefore call 
the distance C A of the right angled triangle the tangent, and 
C B the radius. Now, in the table of tangents for 27° 30' we 
find the decimal .5205 for the length C A, when C B or 
radius is one inch, and as C A in the figure is just one inch, 
then we know that C B is nearly two inches, or just what 
.5205 will go into 1. Thus, 1 -r- .5205 = 1.921 for the depth 
C B. In the Whitworth threads 1/6 is rounded off on the top 
and bottom of the threads, so that we will have to deduct y Q 
of this full depth; thus, 1.921 — J£ = 1.281 inches for one 
side, as shown in the figure. 

In Figure 37, which is a departure from the original in- 
tention of confining to right angled triangles, I have shown a 
class of work that you may be called upon to perform at any 
time, that is, to key a pair of levers on a shaft so that they 
wrill stand at an angle of 80° to each other (or anything 
similar). 

Now, if the angle A B D is 80°, then we can easily see 
that by making the small acute angle B A C we will have a 
right angled triangle, A C D. The object now is to find the 
length of A D (between centers). If we know this distance, 
by placing a pin in each of the holes A D we can make an 
end gauge by means of the micrometer caliper or vernier, and 
get it almost exact to the proper angle. 

We know the lengths A B and D B, but as we are now 
confined to a right angled triangle we must know the distances 
A C and DC, and to do this we must calculate the sides 
AC and B C of the right angled triangle B A C. As the 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 45 




46 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 

longest side, A B, is given as 4", then B C is sine, and C A 
cosine of the angle of 10°. In the table of sines we find the 
number .1736 opposite 10°, then multiplying thus, .1736 X 4 
the radius = .694, the distance B to C; and in the table of co- 
sines we find the number .985 nearly which, multiplied by 
the radius (A B) 4" = 3.94 as the distarce A C. 

The distance D C is now 5J£" — .694 = 4.806 inches. 

The length A C is 3.94 inches, as before explained ; we 
now have a right angled triangle, but we do not know what 
either of the acute angles are, and we can call DC or A C 
radius; one of these sides will be radius and the other tangent. 
Let us call D C radius, then A C is tangent to the angle 
ADC. 

We now divide the tangent A C (or 3.94) by the radius 
D C, or 4.806, which = .8198, or the tangent to the radius of 
1 inch, and in the table of tangents we find this number to 
correspond with the angle 39° 21'. 

Having found the angle, we now find the secant of 39° 21', 
which is 1.2932, which, multiplied by the radius 4.806 = 
6.214 + , the answer, or distance between centers D A. 

The following is another method, provided great accuracy 
is not required. 

In the same example, where two levers are to be keyed on 
a shaft, let Figure 38 represent the end of the shaft, which has 
a fine line lengthwise, say at B, and we want to make another 
line lengthwise at D, that will be just 80 degrees from B. 

The object now is to find the exact distance in a straight 
line from B to D in order to get the line lengthwise on the 
shaft at D. 

As the shaft is 2% 7/ in diameter, then in the angle A C B 
the radius is l%" ', and A B is the sine of the angle (40°.) 

The sine of 40° = .6428, and multiplied by the radius V/i 
= .8033, or the distance A B in a straight line; twice this = 
1.60C6 inches, or the distance required D B. Having center 
lines on the two levers, and using a good new scale and a mag- 
nifying glass, very good results can be obtained in this man- 
ner. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 47 







VD"> 



o y 



<s 



s 




-r-^ 



v* 



48 THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR, 

Figure 40 is similar to Figure 37, except that in the latter 
the two levers form au acute angle, while in the former 
(Figure 40) they form an obtuse angle. Suppose these two 
levers are to be keyed on the shaft F and at an angle of 120 
degrees, as shown. How shall we proceed to get this accu- 
rate ? We will first get a surface plate and level up one of the 
levers (A) by means of parallels, so that a line through the 
center, C D, will be parallel with the surface plate, as shown, 
And as 180° form a half circle, we know that the lever, B, 
should be 60° from D, then, as the length of the lever B is 
6J^ /7 (between centers), which is radius to the angle B F G, 
the line B G becomes the sine of that angle. The sine of 
60° = .866, which, multiplied by the radius 6^ = 5.629 inches, 
and, adding this to the height (whatever it may be raised) from 
the center of the shaft to the surface plate, (in this case 2% // ), 
it will make exactly 8.379 inches ; now place a pin in the lever 
at B, 134 inches diameter, and with an end gauge 8.379 less 
one-half the diameter of this pin, or 7.754 inches (total length) 
will be the exact distance between the plate and the pin at B. 
Another way is to take the distance from the side of the shaft 
shown at I by means of a square and finding the distance, 
I B ; now, as F G is the cosine of the angle B F G and 
the cosine of 60° = .5, then the distance F G is one-half the 
length of radius or S^ // , and, added to one-half the diam- 
eter of the shaft, will be 4.500 inches, or the distance from 
the center of B to I, as shown. 

Figure 41 is an example of lathe work that is impossible 
for anyone to do correctly who does not understand plain 
trigonometry, and yet it is very simple, as I will show you, by 
that process. 

Suppose we have a plate and want to turn a groove on 
one side of it just large enough so that eleven (11) balls, each 
one-quarter {%) inch in diameter, will lie in the groove neatly 
or just fill it. How large will it be? 

A great many not accustomed to work of this kind would, 
at first thought, naturally think that the circumference 
through the center of the balls would be eleven one-quarter 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



49 




50 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

inches, but this is a mistake, as a glance at the figure, which 
is greatly enlarged, will show that the balls will touch only on 
the line A C and not on the circle D E. 

In an example of this kind no drawing is necessary. In 
any circle there are 360 degrees and, dividing this by twice the 
number of balls required, we produce a right angled triangle, 
as shown. 

In this example, dividing 360° by twice the number of 
balls (22) = 16° 22 / nearly. 

We now have an angle of 16° 22', and we know the sine to 
be one-half the diameter of the ball, or .125; then, in the table 
of sines and opposite 16° 22', we find the decimal .2817, which 
means that A C would be .2817 to every inch of B C, and 
1 X .125 -f- .2817 = .4437, or the length B C; then twice this 
= .8874, and the diameter of the ball added to this = 1.137 
inches for the outside diameter of the groove. 

Required the outside diameter of a groove that will admit 
12 \y% inch balls without play. 

360° — 24 = 15° for the angle; as the balls are 1J£", then 
the sine of this angle =.750. In the table of sines and oppo- 
site 15° we find the decimal .259; then .750 -h .259 = 2.895 + " 
for the distance from the center of the circle to the center of 
the balls; then, 2.895 X 2 = 5.790 and the diameter of the ball 
added = 7.290 inches, the answer. 

i Figure 42 will show how to find the length of a (cross) 
belt when the size of the pulleys and their center distances are 
known. One side only of the belt is shown, and by drawing a 
line, A B, parallel from B, we form the angle C B A. As 
the distance A B is known we call it radius, and the belt, 
C B, the secant, and A C the tangent of the angle; we know 
the tangent, because it is equal to one -half of the diameter of 
the large pulley added to one-half the diameter of the small 
pulley = 28 inches, and 28 -f- 74 (the radius) = .3784 + = the 
tangent to radius of 1, and in the table of tangents we find this 
number to correspond with the angle 20° 44'; then the secant 
of this angle equals 1.069, which, multiplied by the radius 
(74 7/ ) = 79.121 nearly ; twice this length added to one-half 



THE MACHINIST AXD TOOL MAKER'S INSTRUCTOR. 51 



52 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 

the circumference of both pulleys = 246,206 inches for the 
total length of the belt. It will be seen from the figure that 
the belt does not meet at the center of the pulleys, B C, but 
this slight difference need not be noticed. 

Figure 43 is similar to Figure 42, except that the former 
is an open belt, but otherwise the dimensions are found in the 
same manner. Thus, the dotted line, B C, is parallel to the 
centers of the two pulleys and is the radius to the angle 
C B A; the difference between half the diameters of the pul- 
leys equals the tangent, which is 8"; then 8 //f -r- 74" = .1081 
or the tangent to a radius of 1 inch, and in the table we find 
this number to correspond to the angle 6° 10', and the secant 
of this angle = 1.0058, which, multiplied by the radius 74 = 
74.429 for the length of one side or A B; then twice this 
length, added to one-half the circumference of both pulleys 
=236.8 inches for the answer, or total length of the belt. 

In nearly all our modern machine tools the cones on the 
feed shafts, that are usually but a short distance from the cone 
on the main spindle, have their steps turned, as shown in 
Figure 44; that is, each step is made X"» %" y Y%' > or un i- 
formly increasing in diameters, and it can easily be seen by 
the figure that if the belt is changed from the position shown 
to the center of the cones that it will be too slack. 

The dimensions of the cones are each 4, 6 and 8 inches for 
the diameters of their steps. We will now find the difference 
in the length of belt required for the center speed and the end 
of the cones. The difference between one half the diameters 
of the smallest step and one-half the diameter of the largest 
step is 2 inches, as shown; then 2'' -4- 16" = .125", or the tan- 
gent of the angle ABC, which corresponds to the angle 
7° 8' ; the secant of this angle, as per table, is 1.0078, 
which, multiplied by the radius (16 r/ ) = 16.124'' for one side 
or A B, and twice this = 32.250 inches nearly, or about one- 
fourth inch too long fur the center speeds. On long belts this 
little difference would not make any trouble, but when they 
are very short they should be made so that the tension of the 
belt will be the same throughout. It is very easy to see that 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



53 




1 




6° 10 , 


i \ 

1 • .s 


: *°"\ J 


4^*9 




36" | 3 
i 
i j 


w _ . 


M".. 












i 




54 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

in this case (Figure 44) that one-half of the circumference of 
both center steps ought to be exactly .125 of an inch longer, or 
.250'' in the whole circumference greater than shown in the 
figure. Thus, the circumference Of 6 /7 = 18.849, and adding 
.250 we have 19.1 nearly, and dividing by 3.1416 = 6.079 // , or 
about 6g 5 ? for the diameter. 

Figure 45 represents the same dimensions of cone and dis- 
tances as in Figure 41, but in the former we have a cross belt; 
it can readily be seen that wherever the belt is placed that the 
angles are the same; when the diameters are uniform as shown, 
the radius of the angles are 16 inches and the tangents are 
each 6 inches; then 6 -f- 16 = .375, or the tangent to radius of 
one inch, which in the table corresponds with the angle 
20° 33 / , and the secant of this angle = 1.068 nearly, which, 
multiplied by the radius 16' f = 17.086 // for the length of one 
side, as A B, etc.; then twice this length, added to one-half 
the circumference of any pair of steps, as shown, will be the 
total length of belt, or about 53.02 7/ , the answer. 

This answer will not be strictly correct, for the reason 
(shown in diagram) that the belt will not be in a straight line 
to the center of the steps and, consequently, it will be a little 
longer than the figures given, but this need not be taken into 
consideration, except when the differences of their diameters 
are very great, which seldom happens in practice. 

It sometimes happens that in an example like Figure 44 
that the middle steps are not of the same diameter; thus, sup- 
pose we have .250 // to be divided proportionately between the 
center steps of two cone pulleys, the diameter of which are 8 
and 10 inches respectively; then 8 4- 10 = 18, and one should 
have 8/18 and the other 10/18, or, what is the same thing, 4/9 
and 5/9 each of the .250 // . Generally it would not make any 
difference if the whole amount were to be added to either one 
of the pair. It will be seen from Figure 45 that in making 
cone pulleys for similar purposes shown that the diameters of 
their steps may increase uniformly if cross belts are to be 
used, but not for open belts, to secure good results. 

In making jigs and templates it is sometimes necessary 
(in order to obtain good results) that we know the distances in 
straight lines and in different directions between holes, etc. 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 00 







A ^ / // 





■** 



In Figure 46 I have shown a template in which three sets 
of holes, A B D, are to be drilled and reamed in a flange 
(nine holes in all); the flange has a hub 4 inches diameter, and 



56 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

the template is bored to suit this hub and is fitted with a bind- 
ing screw, as shown, to hold it securely in any position. 

As there are to be three sets of holes, A, B, D, then after 
the first set is finished we will have to slack the binding 
screw and turn the template exactly 120°. But in order to 
know when it is exactly at that angle and at the same time to 
hold it in that position, we will make an extra hole (C); then, 
after the holes, A B D, are completed in the flange, a pin in 
the template at C is fitted to the hole in the flange at A, and 
it is then ready to drill the next three holes, etc. 

This is very easy to do when the template is at hand, but 
the first thing to do is to make the template and to know how 
to make it correctly ; it makes no difference what hole we 
start from to get the 120°. If we start from B we will have to 
make the template larger, and if we start from D we would 
have to build up on the template above A and to the left of 
it; by starting from A, as shown, we have less to add to the 
template than from any other place. Now, in boring holes in 
jigs and templates, if not too large, they should always be 
done in the lathe, if accuracy is required, because it is almost 
impossible in any drill press to bore a hole and find it true (at 
right angles to the surface of the plate being drilled or bored), 
and even when the drill press is in good condition when new, 
after using it for some time it will get out of shape. But, 
wherever done, in the lathe or drill press, a piece of cast iron 
(or something similar) should be turned four inches diameter 
to fit the template at F, and a plug to fit the spindle in place 
of the center, projecting outside of the face plate an inch or 
two, and turned true and of some exact size; for convenience 
only, suppose this to be one inch, then as the holes, A B G, 
are 8% inches from the center, our block at F, which sup- 
ports the template, can, by means of an end gauge 6.250 inches 
long placed between them, set the template in proper position 
1o get this radius of S% // correct. This is the first operation; 
the template should in the meantime have these holes laid off 
by dividers, etc. roughly (so that we may know about where 
to start the first hole). We will now drill and bore the hole 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 57 

at A. Never depend upon a drill to finish holes in tools of 
this kind, and if these holes are to be tapped, never depend on 
an ordinary tap to run straight, because they never will. 
Special taps with a pilot on the end that fits the hole closely 
should be used, or the thread should be finished or very 
nearly so with a tool in the tool post. If the thread is roughed 
out so that the tap will about enter, then it may be finished in 
this way, but the less you take out with the tap the better, 
and finishing with a tool is the best of all. The hole, A, is 
now supposed to be finished and, before moving the template, 
fit a plug neatly in it (A). The distance in a straight line, 
A B, or the chord A B of 40° is found by doubling the sine 
of half that angle ; thus, the sine of 20° = .342, and multiplied 
by the radius = .342 X 8.75 = 2.9925, and twice this = 5.985, 
or the distance between the centers, A B. If these two holes 
are to be the same size we will make an end gauge 5.985 inches 
long, and placing one end against the plug in A and the other 
end towards H., we will secure a block firmly on the faceplate 
of the lathe and, touching this end gauge near H, now take 
away the end gauge and swing the template so that the plug 
at A will just touch the block near H, and bore the hole B. 

If the template is %" thick, then the block will have to 
be raised high enough to let it pass under. 

The hole C can also be bored in a similar manner. The 
distance, C B, in a straight line is found in the following 
manner: As A C is 120° and A B 40°, then B C is the 
difference between these two angles, or 80°, and the sine of 
40° = .643, which, multiplied by the radius (%%) = 5.626, and 
twice this = the chord of 80° = 11.252" for the distance B C, 
as shown. The hole D is 10° from C and is 10#" from the 
center. We will move the center piece upon which the tem- 
plate is held at F and set it by an end gauge as before. We 
will now find the distance in a straight line from the center of 
C to G, which is on the line D F; then, as C F is the long- 
est line of the angle G F C and is 8% inches long, it is the 
radius of this angle, andG C is the sine; as the sine of 10° = 



58 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 59 

.1736, then multiplied by the radius {S^Q = 1.519 inches for 
the answer. 

A better way, perhaps, for locating the distances in this 
case would be to place the plug in the tail stock spindle (made 
to some standard size), as described, for the reason that it can 
always be moved out of the way, or brought up to the work 
just as close as occasion may require; the end gauge may then 
be used between this and a piece fitted in one of the holes 
already made, for the next operation, etc. 

It is well known that close, accurate work cannot be done 
by the most skillful hand without accurate measuring tools, 
as, for instance, in making the end gauges just mentioned I 
would recommend that they be made of ^g /7 or Jo" round 
steel. Drill rods are the most convenient for this purpose, 
and an instrument that will measure 6 inches will be suffi- 
cient for most purposes, as we can sometimes use two or three 
pieces together to get the necessary length required. 

In figure 46, B represents a bush, sometimes used for 
these templates. It is sometimes more convenient to have 
two or more of them fitted to the same hole, and by using 
threaded bushes (of standard sizes) we are enabled to use one 
with a plain hole, or we may want one with a tapped hole. 
The latter is far the better when tapping holes, and in this 
case the bushes should be made of extra length, and special 
taps would also wear better if made longer in the thread, and 
fluted for as short a distance as possible ; the extra length of 
the bush will guide the tap better and will also prevent its 
wearing away so rapidly; they may be used without temper- 
ing. 

Figure 47 represents a jig made for shaping the piece D, 
and also for drilling it, the surface shaded only being finished, 
including the two edges to form the angle of 37°. 

The plate A is held in the vise of the shaper, while B, on 
which D is clamped, is free to swivel on A. The three succes- 
sive positions, two to shape, and the third to drill (the last 
being the second operation after they are all shaped) are 
shown by the three holes I, I, I, in which a taper pin is used 



60 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

to hold the plate B in the proper position on A; there is also a 
pin B, with a large head, as shown, on which B swivels, and 
this pin is forced down hard by the screw G, the point of 
which bears on one side only of a V groove; this binds the two 
plates B and A firmly together after the dowel-pin is in place; 
the manner of holding D is shown by the screw F pressing it 
against the two pins H H, and also the clamp, as shown. The 
plate C is raised high enough to let D pass under to be drilled. 

The work, D, as shown, is in the proper position for fin- 
ishing the edge, shown at J. In taking the finishing cut an 
end gauge, one inch long, is placed against the finished sur- 
face L, and the tool set so as to touch the other end of this 
gauge. After this operation the taper pin is withdrawn and B 
is swiveled to the next hole I; while in this position the shaded 
surface is finished as well as the edge, shown at O. 

Now, in boring these holes in the plate A, it should be 
swung from a pin on the face plate through the hole K, and 
the holes should be bored with a tool in the tool post, using 
the taper attachment; a taper plug should be used for sizing 
the holes. After both plates are bored in this manner, and 
put together, they may, by being careful, be slightly reamed 
to suit the taper pin. 

If we know the distance in a straight line between these 
holes, by means of gauges, as before stated, we can, with very 
little trouble, make the jig accurate enough for most any pur- 
pose. As the angle calls for 37°, we first find the sine of 18° 
30', which is .317, then .317 X 4%" = 1.5057, and twice this 
= 3.0114 or the distance I' I"; and if the hole through C is 
parallel with E, as shown, and the hole through D is to be in 
the center, then F" C, should be 18° 30'. The drawing is 
wrong, for it shows 37°, but the figures are right, and the angle 
I" F" or the opening between these lines will be the differ- 
ence between 90° and 18° 30' which is 71° 30'; the sine of 
half this angle which is 35° 45' = .584, and multiplied by the 
radius (4%) = 2.775, then twice this sum = 5.550 for the 
answer, as shown. 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 



61 



Figure 48 shows a method of finding the size of a pulley 
(or anything similar) from a piece of the broken rim A B. 
Place the rim on a piece of paper and make the arc of a circle, 
as shown, A B, and in this arc make three or more circles, and 
at the points of intersection of these circles draw lines, and 
the point C, where these lines cross, is the center of the 
pulley, etc. 




62 THE MACHINIST AND TOCXL MAKER'S INSTRUCTOR. 



Figure 49 shows a method of setting a machine at right 
angles to the main shaft. 




D 



9/ 8 49 




Drop a plumb line from the main shaft to the floor, and 
make a mark, D; from ten to twenty feet drop the line again 
and make another mark, K; with a good chalk line make the 
line D B on the floor; from two points on this line F G (any- 
where to suit) make the arcs H H, I I, and, where these lines 
cross each other draw a line, then this line will be at right 
angles to the line D E, as shown. 

Figure 50 represents a plate that is to have a number of 
holes bored through it, and two of these holes should be ex- 
actly six inches center to center, as shown, so that a pair of 
gears will run neatly on studs that are fitted to these holes. 
How shall we do this and know it to be correct? 



THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. 




<£»ye--ty 



<=>--t- 






_.x_. 

/ 1 

/ ! 



4 

x I 



% 






CO 



64 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

We have only to draw the line A B, and we have a right 
angled triangle CBD, the radins of which is six inches, and 
C D is the sine of the angle; dividing 3J4 by 6 we have the 
decimal .5416 which equals the sine to a radius of one inch. 
In the table of sines we fmd this number to correspond 
with the angle 32° 48', nearly; then as C D is the sine, E T> 
becomes cosine to this angle, and the cosine of the angle 32° 
48', as per table, is .8406, and multiplied by the radius (6") 
= 5.0436 inches, as shown. 

Now, if we know these dimensions, by clamping parallels 
F G either on the drill press table or the face plate of a lathe, 
and then letting the plate rest on the parallel F, with an end 
gauge 5.043'' long placed between the end and the parallel 
G, it will be in the proper position for boring the hole C; then 
move the plate against the parallel G, and with two end gauges 
made of good sized wire (or another parallel) placed between 
the plate and the parallel F, just %}4? f l° n g> it will then be in 
the proper position for boring the hole, shown at E, etc. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



65 



NATURAL, SINKS, TANGENTS, COSINES 
AND SECANTS. 







Deg. 








1 DEG. 




MIS 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE . 


SEC. 



2 

4 


.0 

.00058 
.00116 


.0 

.0C058 
.00116 


1. 
1. 
1. 


1. 
1. 
1. 



2 

4 


.01745 
.01803 
.01861 


.01745 
.01803 
.01862 


.99984 
.99984 
.99983 


1.0001 
10001 
1.0002 


6 

8 
10 


.00174 
.00233 
.00291 


.00174 
.00233 
.00291 


.99999 
.99999 
.99999 


1. 
1. 
1. 


6 

8 

10 


.01919 
.01978 
.02036 


.01920 
.01978 
.02036 


.99982 
.999-50 
.99979 


1.0002 
1.0002 
1.0002 


12 
14 
16 


.00349 
.00407 
.00465 


.00349 
.00407 
.00465 


.99999 
.99999 
.99999 


1. 
1. 
1. 


12 

14 
16 


.02094 
.02152 
.02210 


.02095 
.02163 
.02211 


.99978 
.99977 
.99975 


1.0002 
1.0002 
1.0002 


18 
20 
22 


.00524 
.00582 
.00640 


.00524 
.00582 
.00640 


.99998 
.99998 
.99998 


1. 
1. 
1. 


18 
20 
22 


.02269 
.02327 
.02385 


.02269 
.02327 
.02386 


.99974 
.99973 
.99971 


1.0002 
1.0003 
1.0003 


24 ! .00698 
26 .00756 
28 , .00814 


.00698 
.0U756 
.00814 


.99997 
,99997 
.99997 


1. 
1. 
1. 


24 
26 
28 


.02443 

.02501 
.02560 


.02444 
.02502 
.02560 


.99970 
.99969 
.99967 


1.0003 

1.0003 
1.0003 


30 
32 
34 


.00872 
.00931 
.00989 


.00872 
.00931 
.00989 


.99996 
.99996 
.99995 


1. 
1. 
1. 


30 
32 
34 


.02617 
.02676 
.02734 


.02618 
.02677 
.02735 


.99966 
.99964 
.99962 


1.0003 
1.0U03 
1.0004 


36 

38 
40 


.01047 
.01105 
.01163 


.01047 
.01105 
.01164 


.99995 
.99994 
.99993 


1.0001 
1.0001 
1.0001 


36 
38 
40 


.02792 
.02850 
.02908 


.02793 
.02851 
.02909 


.99961 
.99959 
.99958 


1.0004 
1.0004 
1.0004 


42 
44 
46 


.01222 
.01279 
.01338 


.01222 
.01280 
.01338 


.99992 
.99992 
.99991 


1.0001 
1.0001 
1.0001 


42 
44 
46 


.02966 
.03025 
.03083 


.02968 
.03026 
.03084 


.99956 
.99954 
.99952 


1.0004 
1.0004 
1.0005 


48 
50 
52 


.01396 
.01454 
.01513 


.01396 
.01454 
.U1513 


.99990 

.99989 
.99988 


1.0001 
1.0001 
1.0001 


48 
50 
52 


.03141 
.03199 
.032o7 


.03142 
.03201 
.03259 


.99951 
.99949 
.99947 


1.0005 
1.0005 
1.0005 


54 
56 
58 


.01571 
.01629 
.01687 


.01571 

.01629 
.01687 


.99988 
.99987 
.99986 


1.0001 
1.0001 
1.0001 


54 
56 
58 


.03315 
.03374 
.03132 


.03317 
.03375 
.03434 


.99945 
.99943 
.99941 


1.0005 
1.0006 
1.0006 



66 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







2 Deg. 








3 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.08489 
.03548 
.03606 


.03492 
.03»50 
.03608 


.99939 

.99937 
.99935 


1.0006 

1.0006 
1.0006 



2 

4 


.05233 
.05292 
.05349 


.05241 
.05299 
.05357 


.99863 
.99859 
.99857 


1.0014 
1.0014 
1.0014 


6 

8 

10 


.03664 
03722 
.03780 


.03667 
.03725 
.03783 


.99934 
.99931 
.99928 


1.0007 
1.0007 
1.0007 


6 

8 

10 


.05408 
.05466 
.05524 


.05416 
.05474 
.05532 


.99853 

.99850 
.99847 


1.0015 
1.0015 
1.0015 


12 
14 
16 


.03839 

.03897 
.03955 


.03841 
.03899 
•03y58 


.99926 
.99924 
.99922 


1.0007 
1.0008 
1.0008 


12 
14 
16 


.05582 
.05640 
.05698 


.05591 
.05649 
.05707 


.99844 
.99841 
.99837 


1.0016 
1.0016 
1.0016 


18 
20 
22 


.04013 
.04071 
.04129 


.04016 
.04074 
.04133 


.99919 
.99917 
.99915 


1.0008 
1.0008 
1.0008 


18 
20 
22 


.05756 
.05814 
.05872 


.05766 

.05824 
.05883 


.99834 
.99831 
.99827 


1.0017 
1.0017 
1.0017 


24 
26 

28 


.04187 
.04245 
.04304 


.04191 
.04249 
.04308 


.99912 
.99909 
.99907 


1.0009 
1.0009 
1.0009 


24 
26 
28 


.05930 
.05989 
.06047 


.05941 
.05999 
.06058 


.99824 
.99820 
.99817 


1.0018 

1.0018 
1.0018 


30 
32 
34 


.04362 
.04420 
.04478 


.04366 
.04424 
.04482 


.99905 
.99902 
.99900 


1.0009 
1.0010 
1.0010 


30 
32 

34 


.06104 
.06163 
.06221 


.06116 

.06174 
.06233 


.99813 

.99811* 
.99806 


1.0019 
1.0019 
1.0019 


36 

38 
40 


.04536 
.04594 
.04652 


.04541 

.C4599 
.04657 


.99897 
.99894 
.99892 


1.0010 
1.0010 
1.0011 


36 
38 
40 


.06279 
.06337 
.06395 


.06291 
.06349 
.06408 


.99803 
.99799 
.99795 


1.0020 
1.0020 
1.0020 


42 
44 
46 


.04711 

.04768 

.04827 


.04716 

.04774 
.04832 


.99889 
.99886 
.99883 


1.0011 

1.0011 
1.0012 


42 
44 

48 


.06453 

.06511 
.06569 


.06467 
.06525 
.06583 


.99791 
.99788 
.99784 


1.0021 
1.0021 
1.0022 


48 
50 
52 


.04885 
.04943 
.05001 


.04891 
.04949 
.05007 


.99880 

.99877 
.99875 


1.0012 
1.0012 
1.0012 


48 
50 
52 


.06627 
.06685 
.06743 


.06642 
.06700 
.06759 


.99780 
.99776 
.99772 


1.0022 
1.0022 
1.0023 


54 
56 
58 


.05059 
.05117 
.05175 


.05065 
.05124 
.05182 


.99872 
.99869 
.99866 


1.0013 
1.0013 
1.0013 


54 
56 

58 


.06801 
.06859 
.06917 


.06817 
.06876 
.06934 


.99768 
.99764 
.99760 


1.0023 
1.0024 
1.0024 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



67 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







4 De 


G. 




5 Deg. 


MIX 


SINE. 


TANG. 


COSINE. 


S^C. 


MIX 


SINE. 


TANG- 


COSINE- 


SEC. 



2 

4 


.06975 
.07034 
.07091 


.06992 
.07051 
.07109 


.99756 
.99752 
.99748 


1.0024 
1.0025 
1.0025 



2 

4 


.08715 
.08773 
.08831 


.08749 
.08807 
.08866 


.99619 
.99614 
.99609 


1.0038 
1.0039 
1.0039 


6 

8 

10 


.07150 

.07208 
.07266 


.07168 
.07226 
.07285 


.99744 
.99740 
.99736 


1.0026 
1.0026 
1.0026 


6 

8 

10 


.08889 
.08947 
.09005 


.08925 
.08983 
.09042 


.99604 
.99599 
.99594 


1.0040 
1.0040 
1.0041 


12 
14 
16 


.07324 
.07382 
,07440 


.07343 
.07402 
.07460 


.99731 

.99727 
.99723 


1.0027 
1.0027 
1.0027 


12 
14 
16 


.09063 
.09121 
.09179 


.09100 
.09159 
.09218 


.99588 
.99583 
.99578 


1.0041 
1.0042 
1.0042 


18 
20 
22 


.07498 
.07556 
.07614 


.07519 
.07577 
.07636 


.99718 
.99714 
.99709 


1.0028 
1.0029 

1.0029 


18 
20 
22 


.09237 
.09295 
.09353 


.09276 
.09335 
.09394 


.99572 
.99567 
.99562 


1.0043 
1.0043 
1.0044 


24 
26 

28 


.07672 
/TOO 
.07788 


.07694 
.07753 
.07811 


.99705 
.99701 
.99696 


1.0029 
1.0030 
3.0030 


24 
26 
28 


.09411 
.09468 
.09526 


.09453 
.09511 
.09570 


.99556 
.99551 
.99545 


1.0044 
1.0045 
1.0046 


30 
32 

34 


.07846 
.07904 
.07962 


.07870 
.07929 
.07987 


.99692 
.99687 
.99682 


1.0031 
1.0031 
1.0032 


30 
32 
34 


.09584 
.09642 
.09700 


.09629 
.09687 
.09746 


.99539 
.99534 
.99528 


1.0046 
1.0047 
1.0047 


36 
38 
40 


.08020 
.08078 
.08136 


.03076 
.08104 
.08163 


.99678 
.99673 
.99668 


1.0032 
1.0033 
1.0033 


36 

38 
40 


.09758 
.09816 
.09874 


.09805 
.09863 
.09922 


.99523 
.99517 
.99511 


1.0048 
1.0048 
1.0049 


42 
44 
46 


.08194 
.08252 
.08310 


.08221 
.08280 
.08338 


.99664 
.996-^9 
.99654 


1.0034 
1.0034 
1.0035 


42 
44 
46 


.09932 
.09990 
.10047 


.09931 
.10040 
.10099 


.99505 
.99499 

.99494 


1.0050 
1.0050 
1.0051 


48 
50 
52 


.08368 
.08426 

.08484 


.08397 
.08456 
.08514 


.99649 
.99644 
.99639 


1.0035 
1.0036 
1.0036 


48 
50 
52 


.10105 
.10163 
.10221 


.10157 
.10216 
.10275 


,99488 
.99482 
.99476 


1.0051 

1 .0052 
1.0053 


54 
56 

58 


.08542 
.08599 
.08657 


.08573 
.08632 
.08690 


.99634 
.99629 
.99624 


1.0037 
1.0037 
1.0038 


54 
56 
58 


.10279 
.10337 
.10395 


.10334 
.10393 
.10451 


.99470 
.99464 

.99458 


1.0053 
1.0054 
1.0054 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



6 Deg. 


7 Deg. 


MIN 


SINE. 


TANG- 


COSINE. 


SEC- 


MIN 

j 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.10453 
.10511 
.10568 


.10510 

.10569 
.10628 


.99452 
.99446 
.99440 


1.0055 
1.0056 
1.0056 



2 

4 


.12187 
.12245 
.12302 


.12278 
.12337 
.12396 


.99254 
.99247 
.99240 


1.0075 
1.00', 6 
1.0076 


6 

8 
10 


.10626 
.10684 
.10742 


.10687 
.10746 
.10804 


.99434 
.99427 
.99421 


1 0057 
1.0057 
1.0058 


6 

8 

10 


.12360 
.12418 
.12475 


.12455 
.12515 
.12574 


.99233 
.99226 
.99219 


1.0077 
1.0078 
1.0079 


12 
14 
16 


.10800 

.10857 
.10915 


.10863 
.10922 
.10981 


.99415 
.99409 
.99402 


1.0059 
1.0059 
1.0060 


12 
14 
16 


.12533 
.12591 
.12648 


.12633 
.12692 
.12751 


.99211 
.99204 
.99197 


1.0079 
1.0080 
1.0081 


18 
20 
22 


.10973 
.11031 
.11089 


.11040 
.11099 
.11158 


.99396 
.99389 
.99383 


1.0060 
1.0061 
1.0062 


18 
20 
22 


.12706 
.12764 
.12822 


.12810 
.12869 
.12928 


.99189 
.99182 
.99174 


1.0082 
1.0082 
1.0083 


24 
26 

28 


.11147 
.11205 
.11262 


.11217 
.11276 
.11334 


.99377 
.99370 
.9^363 


1.0063 
1.0063 
1.0064 


24 
26 
28 


.12879 
.12937 
.12995 


.12987 
.13045 
.13106 


.99167 
.99169 
.99162 


1.0084 
1.0085 
1.0085 


30 
32 
34 


.11320 
.11378 
.11436 


.11393 
.11452 
.11511 


.99357 
.99350 
.99344 


1.0065 
1.0065 
1.0066 


30 
32 

34 


.13052 
.13110 
.13168 


.13165 
.13224 
.13283 


.99144 
.99137 
.99129 


1.0086 
1.0087 
1.0088 


36 
38 

40 


.11494 
.11551 
.11609 


.11570 
.11629 
.11688 


.99337 
.99331 
.99324 


1.0066 
1.0067 
1.0068 


36 

38 
40 


.13225 
.13283 
.13341 


.13342 
.13402 
.13461 


.99121 
.99114 
.99106 


1.0089 
1.0089 
1.0090 


42 
44 
46 


.11667 
.11725 

.11782 


.11747 
.11806 
.11865 


.99317 
.99310 
.99303 


1.0069 
1.0069 
10070 


42 
44 
46 


.13398 
.13456 
.13514 


.13520 

.13580 
.13639 


.99098 
.93090 
.99082 


1.0091 
1.0092 
1.0092 


48 
50 
52 


.11840 

.11898 
.11956 


.11924 
.11983 
.12042 


.99296 

.99289 
.99283 


1.0071 
1.0071 
1.0072 


48 
50 
52 


.13571 
.13629 

.13687 


.13698 
.13757 
.13817 


.99075 
.99067 
.99059 


1.0093 
1.0094 
1.0095 


54 
56 
58 


.12014 
.12071 
.12129 


.12101 
.12160 
.12219 


.99276 
.99269 
.99262 


1.0073 
1.0074 
1.0074 


54 
56 
58 


.13744 
.13802 
.13859 


.13876 
.13935 
.13995 


.99051 
.99043 
,99035 


1.0096 
1.0097 
1.0097 



THE MACHINIST AXD TOOL MAKER'S INSTRUCTOR. 



69 



NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 



8 Deg. 


9 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE- 


SEC. 



2 

4 


.13917 
.13975 
.14032 


.14054 
.14113 
.14172 


.99027 
.99018 
.99010 


1.0098 
1.0099 
1.0100 




2 

4 


.15643 
.15700 
.15758 


.15838 
.15898 
.15957 


.98769 
.98759 
.98750 


1.0125 
1.0125 
1.0126 


6 

8 
10 


.14090 
.14148 
.14205 


.14232 

.14291 
.14351 


.99002 
.98994 
.98986 


1.0101 
1.0102 
1.0102 


6 

8 

10 


.15816 
.15873 
.15930 


.16017 
.16077 
.16136 


.98741 
.98732 
.98723 


1.0127 
1.0128 
1.0129 


12 
14 
16 


.14263 

.14320 
,14378 


.14410 
.14469 
.14529 


.98977 
.98969 
.98961 


1.0103 
1.U104 
1.01U5 


12 

14 
16 


.15988 
.16045 
.16103 


.16196 
.16256 
.16316 


.98713 
.98704 
.98695 


1.0130 
1.0131 
1.0132 


18 
20 
22 


.14435 
.14493 
.14551 


.14588 
.14648 
.14707 


.98952 
.98944 
.98935 


1.0106 
1.0107 
1.0107 


18 
20 
22 


.16160 
.16218 
.16275 


.16375 
.16435 
.16495 


.98685 
.98676 
.98666 


1.0133 
1.0134 
1.0135 


24 
26 

28 


.14608 
.14666 
.14723 


.14766 
.14826 
.14885 


.98927 
.98918 
.98910 


1.0108 
1.0109 
1.0110 


24 
26 
28 


.16332 
.16390 

.16447 


.16555 
.16614 
.16674 


.98657 
.98647 
.98638 


1.0136 
1.0137 
1.0138 


30 
32 

34 


.14781 
.14838 
.14896 


.14945 
.15004 
.15064 


.98901 
.98893 
.98884 


1.0111 
1.0112 
1.0113 


30 
32 
34 


.16504 
.16562 
.16619 


.16734 
.16794 
.16854 


.98628 
.98619 
.98609 


1.0139 
1.0140 
1.0141 


36 

38 
40 


.14953 
.15011 
.15068 


.15123 
.15183 
.15242 


.98875 
.98867 
.98858 


1.0114 
1.0115 
1.0115 


36 

38 
40 


.16677 
.16734 
.16791 


.16913 
.16973 
.17033 


.98599 
.98590 
.98580 


1.0142 
1.0143 
1.0144 


42 

44 
46 


.15126 
.15183 
.15241 


.15302 
.15362 
.15421 


.98849 
.98840 
.98832 


1.0116 
1.0117 
1.0118 


42 
44 
46 


.16849 
.16906 
.16963 


.17093 
.17153 
.17213 


.98570 
.98560 
.9S551 


1.0145 
1.0146 
1.0147 


48 
50 
52 


.15298 
.15356 
.15413 


.15481 
.15540 
.15600 


.98823 
.98814 
.98805 


1.0119 
1.0120 
1.0121 


48 
50 
52 


.17021 
.17078 
.17135 


.17273 
.17333 
.17393 


,98541 
.98531 
.98521 


1.0148 
1.0149 
1.0150 


54 
56 

58 


.15471 
.15528 
.15586 


.15659 
.15719 
.15779 


.98796 
.98787 
.98778 


1.0122 
1.0123 
1.0124 


54 
56 
58 


.17193 
.17250 
.17307 


.17452 
.17512 
.17572 


.98511 
.98501 
.98491 


1.0151 
1.0152 
1.0153 



70 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



10 Deg. 


II Deg. 


MIN 


SINE. 


TANG- 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.17365 
.17422 
.17479 


.17633 
.17693 
.17753 


.98481 
.98471 
.98460 


1.0154 
1.0155 
1.0156 



2 

4 


.19081 
.19138 
.19195 


.19438 
.19498 
.19559 


.98162 
.98151 
.98140 


1.0187 
1.0188 
1.0189 


6 

8 

10 


.17536 
.17594 
.17651 


.17813 
.17873 
.17933 


.98450 
.98440 
.98430 


1.0157 
1.0158 
1.0159 


6 

8 

10 


.19252 
.19309 
.19366 


.19619 
.19679 
.19740 


.98129 
.98118 
.98107 


1.0191 
1.0192 
1.0193 


12 

14 
16 


.17708 
.17766 
.17823 


.17993 
.18053 
.18113 


.98419 
.98409 
.98399 


1.0160 
1.0162 
1.0163 


12 
14 
16 


.19423 
.19480 
.19537 


.19800 
.19861 
.19921 


.98095 
.98084 
.98073 


1.0194 
1.0195 
1.0196 


18 
20 
22 


.17880 
.17937 
.17994 


.18173 
.18233 
.18293 


.98388 
.98378 
.98367 


1.0164 
1.0165 
1.0166 


18 
20 
22 


.19594 
.19652 
.19709 


.19982 
.20042 
.20103 


.98061 
.98050 
.98038 


1.0198 
1.0199 
1.0200 


24 
26 
28 


.18052 
.18109 
.18166 


.18353 
.18413 
.18474 


.98357 
.98347 
.98336 


1.0167 
1.0168 
1.0169 


24 
26 
28 


.19766 
.19823 
.19880 


.20163 

.20224 
.20285 


.98027 
.98015 
.98004 


1.0201 
1.0202 
1.0204 


30 
32 
34 


.18223 

.18281 
.18338 


.18534 
.18594 
.18654 


.98325 
.98315 
.98304 


1.0170 
1.0171 
1.0172 


30 
32 
34 


.19937 
.19994 
.20051 


.20345 
.20406 
.20466 


.97992 
.97981 
.97969 


1.0205 
1.0206 
1.0207 


36 
38 
40 


.18395 
.18452 
.18509 


.18714 
.18775 
.18835 


.98293 
.98283 
.98272 


1.0174 
1.0175 
1.0176 


36 
38 
40 


.20108 
.20165 
.20222 


.20527 
.20587 
.20648 


.97957 
.97946 
.97934 


1.0208 
1.0210 
1.0211 


42 
44 
46 


.18567 
.18624 
.18681 


.18895 
.18955 
.19016 


.98261 
.98250 
.98239 


1.0177 
1.0178 
1 0179 


42 

44 
46 


.20279 
.20336 
.20393 


.20709 
.20769 
.20830 


.97922 
.97910 
.97898 


1.0212 
1.0213 
1.0215 


48 
50 
52 


.18738 
.18795 
.18852 


.19076 
.19136 
.19196 


.98229 
.98218 
.98207 


1.0180 
1.0181 
1.0182 


48 
50 
52 


.20449 

.20506 
.20563 


.20891 
.20952 
.21012 


.97887 
.97875 
.97863 


1.0216 
1.0217 
1.0218 


54 
56 

58 


.18909 
.18967 
.19024 


.19257 
.19317 
.19377 


.98196 
.98185 
.98174 


1.0184 
1.0185 
1.0186 


54 
56 

58 


.20P20 

.20677 
.20734 


.21073 
.21134 
.21195 


.97851 
.97839 
.97827 


1.0220 
1.0221 
1.0222 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



71 



NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 







12 Deg. 








13 Deg. 




MIN 


SINK. 


TANG. 


COSINE. 


SEC 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.20791 
.20848 
.20905 


.21255 
.21316 
.21377 


.97815 

.97802 
.97790 


1.0223 
1.0425 
1.0226 




2 

4 


.22495 
.22552 
.22608 


.23087 
.23148 
.23209 


.97437 
.97424 
.97411 


1.0263 
1.0264 
1.0266 


6 

8 

10 


.20962 
.21019 
.21075 


.21438 
.21499 
.21560 


.97778 
.97766 
.97754 


1.0227 
1.0228 
1.0230 


6 

8 

10 


.22663 
.22722 
.22778 


.23271 
.23332 
.23393 


.97397 
.97384 
.97371 


1.0267 
1.0268 
1.0270 


12 

14 
16 


.21132 
.21189 
.21216 


.21620 
.21681 

.21742 


.97742 
.97729 
.97717 


1.0231 
1.0232 
1.0234 


12 
14 
16 


.22835 
.22892 
.22948 


.23455 
.23516 
.23577 


.97358 
.97345 
.97331 


1.0271 
1.0273 
1.0274 


18 
20 
22 


.21303 
.21360 
.21417 


.21803 
.21864 
.21925 


.97704 
.97692 
.97679 


1.0235 
1.0236 
1.0237 


18 
20 
22 


.23005 
.23062 
.23118 


.23639 
.23700 
.23762 


.97318 
.97304 
.97291 


1.0276 

1.0277 
1.0278 


24 
26 
28 


.21473 
.21530 
.21587 


.21986 
.22047 
.22108 


.97667 
.97654 
.97642 


1.0239 
1.0240 
1.0241 


24 
26 

28 


.23175 
.23231 
.23288 


.23823 
.23885 
.23946 


.97277 
.97264 
.97250 


1.0280 
1.0281 
1.0283 


30 
32 
34 


.21644 
.21700 
.21757 


.22169 
.22230 
.22291 


.97629 
.97617 
.97604 


1.0243 
1.0244 
1.0245 


30 
32 
34 


.23344 
.234U1 
.23458 


.24008 
.24069 
.24131 


.97237 

.97223 
.97209 


1.0284 

1.0285 
1.0287 


36 

38 
40 


.21814 
.21871 
.21928 


.22352 
.22114 
.22475 


.97592 
.97579 
.97566 


1.0247 
1.0248 
1.0249 


36 
38 
40 


.23514 
.23571 

.23627 


.24192 
.24254 
.24316 


.97196 
.97182 
.97168 


1.0288 
1.0290 
1.0291 


42 
44 
46 


.21984 
.22041 
.22098 


.22536 
.22597 
.22658 


.97553 
.97540 
.97528 


1.0251 
1.0252 
1.0253 


42 
44 
46 


.23684 
.23740 
.23797 


.24377 
.24439 
.24500 


.97155 
.97141 
.97127 


1.0293 
1.0294 
1.0296 


48 
50 
52 


.22155 
.22211 
.22268 


.22719 
.22780 
.22842 


.97515 
.97502 
.97489 


1.0255 
1.0256 
1.0257 


48 
50 
52 


.23853 
.23910 
.23966 


.24562 
.24624 
.24686 


.97113 
.97099 
.97085 


1.0297 
1.0299 
1.0300 


54 
56 

58 


.22325 
.22382 
.22438 


.22903 
.22964 
.23025 


.97476 
.97463 
.97450 


1.0259 
1.0260 
1.0262 


54 
56 

58 


.24023 
.24079 
.24136 


.24747 
.24809 
.24871 


.97071 
.97057 
.97043 


1.0302 

1.0303 
1.0305 



THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 



NATURAL, SINKS, TANGENTS, COSINES 
AND SECANTS. 







14 DEG. 








15 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.24192 
.24248 
.24305 


.24933 
.24994 
.25056 


.97029 
.97015 
.97001 


1.0306 
1.0308 
1.0309 



2 

4 


.25882 
.25938 
.2o994 


.26795 
.26857 
.26919 


.96593 
.96577 
.96562 


1.0353 
1.0354 
1.0356 


6 

8 

10 


.24361 
.24418 
.24474 


.25118 
.25180 
.25242 


.96987 
.96973 
.96959 


1.0311 
1.0312 
1.0314 


6 

8 
10 


.26050 
.26106 
.Z6163 


.26982 
.27044 
.27107 


.96547 
.96.532 
.96517 


1.0358 
1.0359 
1.0361 


12 

14 
16 


.24531 

.24587 
.24643 


.25304 
.25366 
.25427 


.96944 
.96930 
.96916 


1.0315 
1.0317 
1.0318 


12 
14 
16 


.26219 
.26275 
.26331 


.27169 
.27232 
.27294 


.96501 
.96486 
.96471 


1.0362 
1.03^4 
1.0366 


18 
20 
22 


.24700 
.24756 
.24812 


.25489 
.25551 
.25613 


.96901 
.968^7 
.96873 


1.0320 
1.0321 
1.0323 


18 
20 
22 


.26387 
.26443 
.26499 


.27357 
.27419 

.27482 


.96456 
.96440 
.96425 


1.0367 
1.0369 
1.0371 


24 
26 
28 


.24869 
.24925 
.24982 


.25675 
.25737 
.25800 


.96858 
.96844 
.96829 


1.0324 
1.0326 
1.0327 


24 
26 

28 


.26555 
.26612 
.26668 


.27544 
.27607 
.27670 


.96409 
.96394 
.96379 


1.0372 
1.0374 
1.0376 


84 


.2*038 
.25094 
.25150 


.25862 
.25924 
.25986 


.96814 
.96800 
.96785 


1.0329 
1.0330 
1.0332 


30 
32 
34 

36 

38 
40 


.26724 
.26780 
.26836 


.27732 

.27795 
.27858 


.96363 
.96347 
.96332 


1.0377 
1.0379 

1.0381 


36 

38 
40 


.25207 
.25263 
.25319 


.26048 
.26110 
.26172 


.96771 
.96756 
.96741 


1.03a3 
1.0335 
1.0337 


.26892 
.2H948 
.27004 


.27920 
.27983 
.28046 


.96316 
.96300 
.96285 


1.0382 
1.0384 
1.0385 


42 
44 
46 


.25376 
.25432 

.25488 


.26234 
.26297 
.26359 


.96727 
.96712 
.96697 


1.0338 
1.0340 
1.0341 


42 
44 
46 


.27060 
.27116 
.27172 


.28108 

.28171 
.28204 


.96269 
.96253 
.96237 


1.0387 
1.0389 
1.0391 


48 
50 
52 


.25544 
.25600 
.25657 


.26421 
.26483 
.26545 


.96682 
.96667 
.96652 


1.0343 
1.0345 
1.0346 


48 
50 
52 


.27228 

.27284 
.27340 


.28297 
.28360 
.28423 


.96222 
.96206 
.96190 


1.0393 
1.0394 
1.0396 


54 
56 
58 


.25713 
.25769 
.25826 


.26608 
.26670 
.26732 


.96637 
.96622 
.96607 


1.0348 
1.0349 
1.0351 


54 
56 

58 


.27396 
.27452 
.27508 


.28486 
.28548 
.28611 


.96174 
.96)58 
.96142 


1.0398 
1.0399 
1.0401 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



73 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



16 Deg. 


17 Deg. 


MIX 


SINE. 


TANG- 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.27564 
.27619 
.27675 


.28674 
.28737 
.28800 


.96126 
.96110 
.96094 


1.0403 
1.0405 
1.0406 




2 

4 


.29237 
.29293 
.29348 


.30573 
.30637 
.30710 


.95630 
.95613 
.95596 


1.0457 
1.0459 
1.0461 


6 

8 

10 


.27731 
.27787 
.27843 


.28863 
.28926 
.28989 


.96078 
.96062 
.96045 


1.0408 
1.0410 
1.0412 


6 

8 

10 


.29404 
.29459 
.29515 


.30764 
.30827 
.30891 


.95579 
.95562 
.95545 


1.0463 
1.0464 
1.0466 


12 
14 
16 


.27899 
.27955 
.28011 


.29052 
.29116 
.29179 


.96029 
.96013 
.95997 


1.0413 
1.0415 
1.0417 


12 
14 
16 


.29571 
.29626 
.29682 


.30955 
.31019 
.31082 


.95528 
.95510 
.95493 


1.0468 
1.0470 
1.0472 


18 
20 
22 


.28066 
.28122 
.28178 


.29242 
.29305 
.29368 


.95980 
.95964 
.95948 


1.0419 
1.0420 
1.0422 


18 
20 
22 


.29737 
.29793 

.29848 


.31146 
.31210 
.31274 


.95476 
.95459 
.95441 


1.0474 
1.0476 
1.0478 


24 
26 
28 


.28234 
.28290 
.28346 


.29431 
.29496 

.29558 


.95931 
.95915 
.95898 


1.0424 
1.0126 
1.0428 


24 
26 
28 


.29904 
.29959 
.30015 


.31338 
.31402 
.31466 


.95424 
.95406 
.95389 


1.0479 
1.0481 
1.0483 


30 
32 
34 


.28401 
.28457 
.28513 


.29621 
.29684 
.29748 


.95882 
.95865 
.95849 


1.0429 
1.0431 
1.0433 


30 
32 
34 


.30070 
.30126 
.30181 


.31530 
.31594 
.31658 


.95372 
.95354 
.95336 


1.0485 
1.0487 
1.0489 


36 

38 
40 


.28569 
.28624 
.28680 


.29811 
.29874 
.29938 


.95832 
.9.5815 
.95799 


1.0435 
1.0437 
1.0438 ; 


36 
38 
40 


.30237 
.30292 
.30348 


.31722 
.31786 
.31850 


.95319 
.95301 
.95284 


1.0491 
1.0493 
1.0495 


42 
44 
46 


.28736 
.28792 
.28847 


.30001 
.30065 
.30128 


.95782 
.95765 
.95749 


1.0440 ! 

1.0442 

10444 


42 
44 
46 


.30403 
.30458 
.30514 


.31914 
.31978 
.32012 


.95266 
.95248 
.95231 


1.0497 . 

1.0499 

1.0501 


48 
50 
52 


.28903 
.28959 
.29014 


.30192 
.30255 
.30318 


.95732 
.95715 
.95698 


1.0446 

1.0448 
1.0449 


48 
50 
52 


.30569 
.30625 
.30680 


.32106 
.32170 
.32235 


.95213 
.95195 
.95177 


1.0503 
1.0505 
1.0507 


54 
56 
58 


.29070 
.29126 
.29181 


.30382 
.30446 
.30509 


.95681 
.95664 
.95647 


1.0451 
1.0453 
1.0455 


54 
56 
58 


.30735 
.30791 
.30846 


.32299 
.32363 
.32427 


.95159 
.95141 
.95123 


1.0509 
1.0511 
1.0513 



74 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



18 Deg. 


19 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.30902 
.30957 
.31012 


.32492 
.32556 
.32620 


.95105 
.95088 
.95069 


1.0515 
1.0517 
1.0519 



2 

4 


.32557 
.3261 2 
.32667 


.34433 
.34498 
.34563 


.94552 
.94533 
.94514 


1.0576 
1.0578 
1.0580 


6 

8 

10 


.31067 
.31123 
.31178 


.32685 
.32749 
.32814 


.95051 
.95033 
.95015 


1.0521 
1.0523 
1.0525 


6 

8 

10 


.32722 

.32777 
.32832 


.34628 
.34693 
.34758 


.94495 
.94476 
.94457 


1.0582 
1.0585 
1.0587 


12 
14 
16 


.31233 

.31289 
,31344 


.32878 
.32943 
.33007 


.94997 
.94979 
.94961 


1.0527 
1.0529 
1.0531 


12 
14 
16 


.32886 
.32941 
.32996 


.34823 
.31889 
.34954 


.94437 
.94418 
.94399 


1.0589 
1.0591 
1.0593 


18 
20 
22 


.31399 
.31454 
.31509 


.33072 
.33136 
.33200 


.94942 
.94924 
.94906 


1.0533 
1.0535 
1.0537 


18 
20 
22 


.33051 
.33106 
.33161 


.35019 
.35085 
.35150 


.94380 
.94361 
.94341 


1.0595 
1.0598 
1.0600 


24 
26 

28 


.31565 
.31620 
.31675 


.33265 
.3334) 
.33395 


.94887 
.94869 
.94851 


1.0539 
1.0541 
1.0543 


24 
26 
28 


.33216 
.33271 
.33326 


.35215 
.35281 
.35346 


.94322 
.94303 
.94283 


1.0602 
1.0604 
1.0606 


30 
32 
34 


.31730 
.31785 
.31841 


.33459 
.33524 
.33589 


.94832 
.94814 
.94795 


1.0545 
1.0547 
1.0549 


30 
32 
34 


.33381 
.33435 
.33490 


.35412 
.35477 
.35543 


.94264 
.94245 
.94225 


1.0608 
1.0611 
1.0613 


36 
38 
40 


.31896 
.31951 
.32006 


.33654 
.33718 
.33783 


.94777 
.94758 
.94739 


1.0551 
1.0553 
1.0555 


36 
38 
40 


.33545 
.33599 
.33655 


.35608 
.35674 
.35739 


.94206 
.9H86 
.94166 


1.0615 
1.0617 
1.0619 


42 
44 
46 


.32061 
.32116 
.32171 


.33848 
.33913 
.33978 


.94721 
.94702 
.94683 


1.0557 
1.0559 
1.0561 


42 
44 
46 


.33709 
.33764 
.33819 


.35805 
.35871 
.35936 


.94147 
.94127 
.94108 


1.0621 
1.0624 
1.0626 


48 
50 
52 


.32226 

.32282 
.32337 


.34043 
.34108 
.34172 


.94665 
.94646 
.94627 


1.0563 
1.0566 
1.0568 


48 
50 
52 


.33874 
.33928 
.33983 


.36002 
.36068 
.36134 


,940^8 
.94068 
.94048 


1.0628 
1.0630 
1.0633 


54 
56 

58 


.32392 
.32447 
.32502 


.34237 
.34302 
.34368 


.94608 
.94589 
.94571 


1.0570 
1.0572 
1.0574 


54 
56 
58 


.34038 
.34092 
.34147 


.36199 
.36265 
.36331 


.94029 
.94009 
.93989 


1.0635 
1.0637 
1.0639 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







20 DEG. 








21 Deg. 




min 


SINK. 


TANG. 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE . 


SEC. 



2 

4 


.34202 
.34257 
.34311 


.36397 
.36463 
.36529 


.93969 
.93949 
.93929 


1.0642 
1.0644 
1.0646 ; 



2 

4 


.35837 
.35891 
.35945 


.38386 
.38453 
.38520 


.93358 
.93337 
.93316 


1.0711 
10714 
1.0716 


6 

8 

10 


.34366 
.34420 
.34475 


.36595 
.36661 

.36727 


.93909 
.93889 
.93869 


1.0648 
1.0651 
1.0653 


6 

8 

10 


.36000 
.36054 
.36108 


.38587 
.386.53 
.38720 


.93295 
.93274 
.93253 


1.0719 
1.0721 
1.0723 


12 

14 
16 


.34530 

.34584 
.34639 


.36793 
.36859 
.36925 


.93849 
.93829 
.93809 


1.0655 
1.0658 1 
1.0660 


12 
14 
16 


.36162 
.36217 
.36271 


.38787 
.38854 
.38921 


.93232 
.93211 
.93190 


1.0726 
1.0728 
1.07a! 


18 
20 
22 


.34693 
.34748 
.34803 


.36991 
.37057 
.37123 


.93789 
.93769 
.93748 


1.0662 i 

1.0664 

1.0667 


18 
20 
22 

24 
26 
28 


.36325 
.36379 
.36433 


.38988 
.39055 
.39122 


.93169 
.93148 
.93127 


1.0733 
1.0736 
1.0738 


24 
26 
28 


.34857 
.34912 
.34966 


.37189 
.37256 
.37322 


.93728 
.93708 
.93687 


1.0669 
1.0672 

1.0o74 


.36488 
.36542 
.36596 


.39189 
.39257 
.39324 


.93105 
.93084 
.93063 


1.0740 
1.0743 
1.07,15 


30 
32 
34 


.35021 
.35075 
.35129 


.37388 
.37455 
.37521 


.93667 
.93647 
.93626 


1.0676 
1.0678 
1.0681 


30 
32 
34 


.36650 
.36704 
.36758 


.39390 
.39458 
.39525 


.93042 
.93020 
.92999 


1.0748 
1.075O 
1.0753 


36 

38 
40 


.35184 
.35238 
.35293 


.37587 
.37654 
.87720 


.93606 
.93585 
.93565 


1.0683 
1.0685 
1.0688 I 


36 

38 
40 


.36812 
.36866 
.36920 


.39593 
.39660 
.39727 


.92977 

.92956 
.929.35 


1.0755 
1.0758 
1.0760 


42 

44 
46 


.35347 
.35402 
.35456 


.37787 
.37853 
.379.0 


.93544 
.93524 
.93503 


1.0690 
1.06y2 
1.0695 


42 
44 
46 


.36975 
.37029 
.37o83 


.39795 
.39862 
.39929 


.92913 

.92892 
.92870 


1.0762 
1.0765 
1.0768 


48 
50 
52 


.35511 
.35565 
.35619 


.37986 
.38053 
.38119 


.93483 
.94462 
.93441 


1.0697 
L.0699 j 
1.0702 ; 


48 
50 
52 


.37137 
.37191 
.37245 


.39997 
.40064 
.40132 


.92848 
.92827 
.92805 


1.0770 
1.0773 
1.0775 


54 
56 

58 


.35674 

.35728 
.35782 


.38186 
.38253 
.383 J 9 


.93420 
.93399 
.93379 


1.0704 

1.0707 
1.0709 


54 
56 

58 


.37299 
.37353 
.37406 


.40199 
.40267 
.40335 


.92783 
.92762 
.92740 


1.0778 
1.0780 
1.0783 



76 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







22 Deg. 








23 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TA*G. 


COSINE. 


SEC. 



2 

4 


.37460 
.37514 
.37568 


.40402 
.40470 
.40538 


.92718 
.92697 
.92674 


1.0785 
1.0788 
1.0790 



2 

4 


.39073 
.39126 
.39180 


.42447 
.42516 

.42585 


.92050 
.92028 
.92005 


1.0864 
1.086^ 
1.086a 


6 

8 

10 


.37622 
.37676 
.37730 


.40606 
.40673 
.40741 


.92653 
.92631 
.92619 


1.0793 
1.0795 
1.0798 


6 

8 

10 


.39234 
.39287 
.39341 


.42653 
.42722 
.42791 


.91982 
.91959 
.91936 


1.0872 
1.0874 
1.0877 


12 
14 
16 


.37784 

.37838 
.37892 


.40809 
.40877 
.40945 


.92587 
.92565 
.92543 


1.0801 
1.0803 
1.0806 


12 
14 
16 


.39394 
.39447 
.39501 


.42860 
.4-929 
.42998 


.91913 

.91890 
.91867 


1.0880 
1.0882 
1.0885 


18 
20 
22 


.37945 
.37999 
.38053 


.41013 
.41081 
.41149 


.92521 
.92499 
.92477 


1.0808 
1.0811 
1.0813 


18 
20 
22 


.39554 
.39608 
.39661 


.43067 
.43136 
.43205 


.91845 
.91821 
.91798 


1.0888 
1.0891 
1.0893 


24 
26 
28 


.38107 

.38161 

. .38214 


.41217 
.41285 
.41353 


.92454 
.92432 
.92410 


1.0816 
1.0819 
1.0821 


24 
26 
28 


.39715 
.39768 
.39821 


.43274 
.43343 
.43412 


.91775 
.917o2 
.91729 


1.0896 
1.0899 
1.0902 


30 
32 
34 


.38268 
.38322 
.38376 


.41421 
.41489 
.41558 


.92388 
.92366 
.92343 


1.0824 
1.0826 
1.0829 


30 
32 
34 


.39875 
.39928 
.39981 


.43481 
.43550 
.43619 


.91706 
.91683 
.91659 


1.0904 
1.0907 
1.0910 


36 
38 
40 


.38429 
.3S483 
.38537 


.41626 
.41694 
.41762 


.92321 
.92298 
.92276 


1.0832 
1.0834 
1.0837 


36 
38 
40 


.40035 
.40088 
.40141 


.43689 
.43758 
.43827 


.91636 
.91613 

.91589 


1.0913 

1.0915 
1.0918 


42 
44 
46 


.38590 
.38*U4 
.38698 


.41831 

.41899 
.41967 


.92254 
.92231 
.92208 


1.0840 
1.0842 
1.0845 


42 
44 
46 

48 
50 
52 


.40195 
.40248 
.40301 


.43S97 
.43966 
.44036 


.91566 
.91543 
.91519 


1.0921 
1.0924 
1.0927 


48 
50 
52 


.38751 
.38805 
.38859 


.42036 
.42104 
.42173 


.92186 
.92164 
.92141 


1.0847 
1.0850 
1.0853 


.40354 
.40407 
.40461 


.44105 
.44175 
.44244 


.91496 
.91472 
.91449 


1.0929 
1.0932 
1.0935 


54 
56 
58 


.38912 
.38966 
.39019 


.42241 
.42310 
.42379 


.92118 
.92096 
.92073 


1.0855 
1.0858 
1.0861 


54 
56 
58 


.40514 
.40567 
.40620 


.44314 
.44383 
.44453 


.91425 
.91402 
.91378 


1.0938 
1.0941 
1.0943 



THE MACHINIST AND TOOX. MAKEE's INSTRUCTOR. 



77 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



24 DEG. 


25 DEG. 


MIN 


SINE. 


TANG- 


COSINE. 


sec. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.40674 
.40727 
.40780 


.44523 
.44592 
.44662 


.91354 
.91331 
.91307 


1.0946 
1.0949 
1.0952 



2 

4 


.42262 
.42314 
.42367 


.46631 
.46701 
.46772 


.90631 
.90606 
.90581 


1.1034 
1.1037 
1.1040 


6 

8 
10 


.40833 
.40886 
.40939 


.44732 
.44802 
.44872 


.91283 
.91259 
.91236 


1.0955 
1.0958 
1.0961 


6 

8 
10 


.42420 
.42472 
.42525 


.46843 
.46914 
.46985 


.90557 
.90532 
.90507 


1.1043 
1.1046 
1.1049 


12 
14 
16 


.40992 
.41045 
.41098 


.44941 

.45012 
.45081 


.91212 
.91188 
.91164 


1.0963 
1.0966 
1.0969 


12 
14 
16 


.42578 
.42630 
.42683 


.47056 
.47127 
.47198 


.90482 
.90458 
.90433 


1.1052 
1.1055 
1.1058 


18 
20 
22 


.41151 
.41204 
.41257 


.45152 
.45221 

.45292 


.91140 
.91116 
.91092 


1.0972 
1.0975 
1.0978 


18 
20 
22 


.42736 
.42788 
.42841 


.47270 
.47341 
.47412 


.90408 
.90383 
.90358 


1.1061 
1.1064 
1.1067 


24 
26 
28 


.41310 
.41363 
.41416 


.45362 
.45432 
.45502 


.91068 
.91044 
.91020 


1.0981 
1.0984 
1.0987 


24 
26 
28 


.42893 
.42946 
.42999 


.47483 
.47555 
.47626 


.90333 
.90308 
.90283 


1.1070 
1.1073 
1.1076 


30 
32 
34 


.41469 
.41522 
.41575 


.45572 
.45643 
.45713 


.90996 
.90972 
.90948 


1.0989 
1.0992 
1.0995 


30 
32 
34 


.43051 
.43103 
.43156 


.47697 
.47769 
.47840 


.90258 
.90233 
.90208 


1.1079 
1.1082 
1.1085 


36 
38 
40 


.41628 
.41681 
.41734 


.45783 
.45854 
.45924 


.90923 
.90899 
.90875 


1.0998 
1.1001 
1.1004 


36 
38 
40 


.43209 
.43261 
.43313 


.47912 
.47983 
.48055 


.90183 
.90158 
.90133 


1.1088 
1.1092 
1.1095 


42 
44 
46 


.41787 
.41839 
.41892 


.45995 
.46065 
.46136 


.90851 
.90826 
.90802 


1.1007 
1J010 
11013 


42 
44 
46 


.43366 
.43418 
.43471 


.48127 
.48198 
.48270 


.90107 
.90082 
.90057 


1.1098 
1.1101 
1.1104 


48 
50 
52 


.41945 
.41998 
.42051 


.46206 
.46277 
.46348 


.90778 
.90753 
.90729 


1.1016 
1.1019 
1.1022 


48 
50 
52 


.43523 
.43575 
.43628 


.48342 
.48413 
.48485 


.9C032 
.90006 
.89981 


1.1107 
1.1110 
1.1113 


54 
56 
58 


.42104 
.42156 
.42209 


.46418 
.46489 
.46560 


.90704 
.90680 
.90655 


1.1025 
1.1028 
1.1031 


54 
56 

58 


.436S0 
.43732 
.43785 


.48557 
.48629 
.48701 


.89956 
.89930 
.89905 


1.1116 
1.1119 
1.1123 



78 



THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



26 Deg. 


27 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.43837 
.43889 
.43941 


.48773 
.48845 
.48917 


.89879 
.89854 
.89828 


1.1126 
1.1129 
1.1132 



2 
4 


.45400 
.45451 
.45503 


.50952 
.51026 
.51099 


.89100 

.89087 
.89048 


1.1223 
1.1226 
1.1229 


6 

8 

10 


.43994 
.44046 
.44098 


.48989 
.49062 
.49134 


.89803 
.89777 
.89751 


1.1135 
1.1138 
1.1142 


6 

8 

10 


.45554 
.45606 
.45658 


.51172 
.51246 
.51319 


.89021 
.88995 
.88968 


1.1233 
1.1237 
1.1240 


12 
14 
16 


.44150 
.44203 
,44255 


.49206 
.49278 
.49351 


.89726 
.89700 
.89674 


1.1145 
1.1148 
1.1151 


12 
14 
16 


.45710 
.45761 
.45813 


.51393 
.51466 
.51540 


.88941 
.88915 
.88888 


1.1243 
1.1247 
1.1250 


18 
20 
22 


.44307 
.44359 
.44411 


.49423 
.49495 
.49568 


.89648 
.89623 
.89597 


1.1155 
1.1158 
1.1161 


18 
20 
22 


.45865 
.45916 
.45968 


.51614 
.51687 
.51761 


.88862 
.88835 
.88808 


1.1253 
1.1257 
1.1260 


24 
26 
28 


.44463 
.44515 
.44568 


.49640 

.49713 

.49785 


.89571 
.89545 
.89519 


1.1164 
1.1167 
1.1171 


24 
26 

28 


.46020 
.46071 
.46123 


.51835 
.51909 
.51983 


.88781 
.88755 
.88728 


1.1264 
1.1267 
1.1270 


30 
32 
34 


.44620 
.44672 
.44724 


.49858 
.49931 
.50003 


.89493 

.89467 
.89441 


1.1174 
1.1177 
1.1180 


30 
32 

34 

36 
38 
40 


o46175 
.46226 
.46278 


.52057 
.52130 
.52204 


.88701 

.88674 
.88647 


1.1274 

1.1277 
1.1281 


36 

38 

40 


.44776 
.44828 
.44880 


.50076 
.50149 

.50222 


.89415 
.89389 
.89363 


1.1184 
1.1187 
1.1190 


.46329 
.46381 
.46433 


.52279 
.52353 
.52427 


.88620 
.88593 
.88566 


1.1284 
1.1287 
1.1291 


42 
44 
46 


.44932 

.14984 
.45036 


.50295 
.50367 
.50440 


.89337 
.89311 
.89285 


1.1193 
1.1197 
1.1200 


42 
44 
46 


.46484 
.46536 
.46587 


.52501 

.52575 
.52649 


.88539 
.88512 
.88485 


1.1294 
1.1298 
1.1301 


48 
50 
52 


.45088 
.45139 
.45191 


.50513 

.50586 
.50660 


.89258 
.89232 
.89206 


1.1203 
1.1207 
1.1210 


48 
50 
52 

54 
56 

58 


.46639 
.46690 
.46741 


.52724 

.52798 
.52873 


,88458 
.88431 
.88404 


1.1305 

1.1308 
1.1312 


54 
56 
58 


.45243 
.45295 
.45347 


.50733 
.50806 
.50879 


.89179 

.89153 
.89127 

- 


1.1213 

1.1217 
1.1220 


.46793 

.46844 
.46896 


.52947 
.53022 
.53096 


.88376 
.88349 
.88322 


1.1315 
1.1319 
1.1322 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



79 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



28 Deg. 


29 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.46947 
.46998 
.47050 


.53171 
.53245 
.53320 


.88295 
.88267 
.88240 


1.1326 
1.1329 
1.1333 



2 

4 


.48481 
.48532 
.48582 


.55431 
.55507 
.55583 


.87462 
.87434 
.87405 


1.1433 
11437 
1.1441 


6 

8 

10 


.47101 
.47152 
.47204 


.53395 
.53470 
.53544 


.88213 
.88185 
.88158 


1.1336 
1.1339 
1.1343 


6 

8 

10 


.48633 
.48684 
.48735 


.55659 
.55735 
.55812 


.87377 
.87349 
.87321 


1.1445 
1.1448 
1.1452 


12 
14 
16 


.47255 
.47306 
.47357 


.53619 
.53694 
.53769 


.88130 
.88103 
.88075 


1.1347 
1.1350 
1.1354 


12 
14 
16 


.48786 
.48836 
.48887 


.55888 
.55964 
.56041 


.87292 
.87264 
.87235 


1.1456 
1.1459 
1.1463 


18 
20 
22 


.47409 
.47460 
.47511 


.53844 
.53919 
.53994 


.88047 
.88020 
.87992 


1.1357 
1.1361 
1.1365 


18 
20 
22 


.48938 
.48989 
.49040 


.56117 
.56194 
.56270 


.87207 
.87178 
.87150 


1.1467 
1.1471 
1.1474 


24 
26 

28 


.47562 
.47613 
.47664 


.54070 
.54145 
.54220 


.87965 
.87937 
.87909 


1.1368 
1.1372 
1.1375 


24 
26 

28 


.49090 
.49141 
.49191 


.56347 
.56424 
.56500 


.87121 
.87093 
.87064 


1.1478 
1.1482 
1.1486 


30 
32 
34 


.47716 

.477*7 
.47818 


.54295 
.54371 
.54446 


.87881 
.87854 
.87826 


1.1379 
1.1382 
1.1386 


30 
32 
34 


.49242 
.49293 
.49343 


.56577 
.56654 
.56731 


.87035 
.87007 
.86978 


1.1489 
1.1493 
1.1497 


36 
38 

40 


.47869 
.47920 
.47971 


.54522 

.54597 
.54673 


.87798 
.87770 
.87742 


1.1390 
1.1393 
1.1397 


36 

38 
40 


.49394 
.49445 
.49495 


.56808 
.56885 
.56962 


.86949 
.86920 
.86892 


1.1502 
1.1505 
1.1508 


42 
44 
46 


.48022 
.48073 
.48124 


.54748 
.54824 
.54899 


.87714 
.87686 
.87659 


1.1401 
1.1404 
1.1408 


42 
44 
46 


.49546 
.49596 
.49647 


.57039 
.57116 
.57193 


.86863 
.86834 
.86805 


1.1512 
1.1516 
1.1520 


48 
50 
52 


.48175 
.48226 

.48277 


.54975 
.55051 
.55127 


.87631 
.87602 
.87574 


1.1411 
L.1415 

1.1419 


48 
50 
52 


.49697 
.49748 
.49798 


.57270 
.57348 
.57425 


.86776 
.86747 
.86718 


1.1524 
1.1528 
1.1531 


54 
56. 

58 


.48328 
.48379 
.48430 


.55203 
.55279 
.55355 


.87546 
.87518 
.87490 


1.1422 
1.1426 
1.1430 


54 
56 
58 


.49849 
.49899 
.49949 


.57502 
.57580 
.57657 


.86689 
.86660 
.86631 


1.1535 
1.1539 
1.1543 



80 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINKS, TANGENTS, COSINES 
AND SECANTS. 







30 Deg. 








31 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.50000 
.50050 
.50100 


.57735 

.57812 
.57890 


.86602 
.86573 
.86544 


1.1547 
1.1551 
1.1555 



2 

4 


.51504 
.51554 
.51603 


.60086 
.60165 
.60244 


.85717 
.85687 
.85657 


1.1666 
1.1670 
1.1674 


6 

8 

10 


.50151 
.50201 
.50252 


.57968 
.58046 
.58123 


.86515 

.86486 
.86456 


1.1559 
1.1562 
1.1566 


6 

8 

10 


.51653 
.51703 
.51753 


.60324 
.60403 
.60482 


.85627 
.85596 
.85566 


1.1678 
1.1683 
1.1687 


12 

14 
16 


.50302 
.50352 
.50402 


.58201 
.58279 
.58357 


.86427 
.86398 
.86369 


1.1570 
1.1574 
1.1578 


12 

14 
16 


.51803 
.51852 
.51902 


.60562 
.60642 
.60721 


.85536 
.85506 
.85476 


1.1691 
1.1695 
1.1699 


18 
20 
22 


.50453 
.50503 
.50553 


.58435 
.58513 
.58591 


.86339 
.86310 
.86281 


1.1582 
1.1586 
1.1590 


18 
20 
22 


.51952 
.52001 
.52051 


.60801 
.60880 
.60960 


.85446 
.85415 

.85385 


1.1703 
1.1707 
1.1712 


24 
26 
28 


.50603 " 

.50653 

.50703 


.58669 
.58748 
.58826 


.86251 
.86222 
.86192 


1.1594 

1.1598 
1.1602 


24 
26 
28 


.52101 
.52150 
.52200 


.61040 
.61120 
.61200 


.85355 
.85325 
.85294 


1.1716 
1.1720 

1.1724 


30 
32 
34 


.50754 
.50804 
.50854 


.58904 
.58983 
.59061 


.86163 
.86133 
.86104 


1.1606 
1.1610 
1.1614 


30 
32 
34 


.52250 
.52300 
.52349 


.61280 
.61360 
.61440 


.85264 
.85233 
.85203 


1.1728 
1.1734 
1.1737 


36 

38 
40 


.50904 
.50954 
.51004 


.59140 
.59218 
.59297 


.86074 
.86044 
.86015 


1.1618 
1.1622 
1.1626 


36 

38 
40 


.52398 
.52448 
.52498 


.61520 
.61600 
.6168J 


.85173 
.85142 
.85112 


1.1741 
1.1745 
1.1749 


42 
44 
46 


.51054 
.51104 
.51154 


.59375 
.59454 
.59533 


.85985 
.85955 
.85925 


1.1630 
1.1634 
1.1638 


42 
44 
46 


.52547 
.52597 
.52646 


.61761 
.61841 
.61922 


.85081 
.85050 
.85020 


1.1753 
1.1758 
1.1762 


48 
50 
52 


.51204 
.51254 
.51304 


.59612 
.59691 
.59770 


.85896 
.85866 
.85836 


1.1642 
1.1646 
1.1650 


48 
50 
52 


.52696 
.52745 
.52794 


.62002 
.6-083 
.62164 


.84989 
.84958 
.849>8 


1.1766 
1.1770 
1.1775 


54 
56 

58 


.51354 
.51404 
.51454 


.59848 
.59928 
.60007 


.85806 
.85776 
.85746 


1.1654 
1.1658 
1.1662 


54 
56 
58 


.52844 
.52893 
.52943 


.62244 
.62325 
.62406 


.84897 
.84866 
.84835 


1.1779 
1.1783 
1.1787 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



81 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 




82 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



34 DEG. 


35 DEG. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.55919 

.55967 
.56016 


.67451 
.67535 
.67620 


.82903 
.82871 
.82838 


1.2062 
1.2067 
1.2072 



2 

4 


.57357 
.57405 
.57453 


.70021 

.70107 
.70194 


.81915 

.81882 
.81848 


1.2208 
1.2213 
1.2218 


6 

8 
10 


.56064 
.56112 
.56160 


.67705 

.67790 
.67875 


.82806 

.82773 
.82741 


1.2076 
1.2081 
1.2086 


6 

8 
10 


.57500 
.57548 
.57596 


.70281 
.70368 
.70455 


.81815 
.81781 
.81748 


1.2223 
1.2228 
1.2233 


12 
14 
16 


.56208 
.56256 
,56304 


.67960 
.68045 
.68130 


.82708 
.82675 
.82642 


1.2091 
1.2095 
1.2100 


12 
14 
16 


.57643 
.57691 

.57738 


.70542 
.70629 
.70716 


.81714 
.81681 
.81647 


1.2238 
1.2243 
1.2248 


18 
20 
22 


.56352 
.56400 
.56449 


.68215 
.68300 
.68386 


.82610 

.82577 
.82544 


1.2105 
1.2110 
1.2115 


18 
20 
22 


.57786 
.57833 
.57881 


.70804 
.70891 
.70979 


.81614 
.81580 
.81546 


1.2253 
1.2258 
1.2263 


24 
26 
28 


.56497 
.56545 
.56593 


.68471 
.68557 
.68642 


.82511 

.82478 
.82445 


1.2119 
1.2124 
3.2129 


24 
26 
28 


.57928 
.57975 
.58023 


.71066 
.71154 
.71241 


.81513 
.81479 
.81445 


1.2268 
1.2273 
1.2278 


30 
32 
34 


.56640 
.56688 
.56736 


.68728 
.68814 
.68900 


.82412 
.82379 
.82346 


1.2134 
1.2139 
1.2144 


30 
32 
34 

36 
38 
40 


.58070 
.58117 
.58165 


.71329 
.71417 
.71505 


.81411 
.81378 
.81344 


1.2283 

1.2288 
1.2293 


36 
38 
40 


.56784 
.56832 
.56880 


.68985 
.69071 
.69157 


.82313 

.82280 
.82247 


1.2149 
1.2153 
1.2158 


.58212 
.58260 
.58307 


.71593 
.71681 
.71769 


.81310 
.81276 
.C1242 


1.2298 
1.2304 
1.2309 


42 
44 
46 


.56928 
.56976 
.57023 


.69243 
.69329 
.69415 


.82214 
.82181 
.82148 


1.2163 
1.2168 
1.2173 


42 
44 
46 


.58354 
.58401 
.58449 


.71857 
.71945 
.72034 


.81208 
.81174 
.81140 


1.2314 
1.2319 
1.2324 


48 
50 
52 


.57071 
.57119 
.57167 


.69502 
.69588 
.69674 


.82115 

.82082 
.82048 


1.2178 
1.2183 
1.2188 


48 
50 
52 


.58496 
.58543 
.58590 


.72122 
.72211 

.72299 


.81106 

.81072 
.81038 


1.2329 
1.2335 
1.2340 


54 
56 

58 


.57214 
.57262 
.57310 


.69761 

.69847 
.69934 


.82015 
.81982 
.81948 


1.2193 
1.2198 
1.2203 


54 
56 
58 


.58637 
.58684 
.58731 


.72388 
.72476 
.72565 


.81004 

.80970 
.80936 


1.2345 
1.2350 
1.2355 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



83 



NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 



1 
36 Deg. 


37 Deg. 


MIS 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.58778 
.58825 
..58872 


.72654 
.72743 

.72832 


.80902 
.80867 
.80833 


1.2361 
1.2366 
1.2371 




2 

4 


.60181 
.60228 
.60274 


.75355 
.75446 
.75538 


.79863 
.79828 
.79793 


1.2521 
12527 
1.2532 


6 

8 
10 


.58919 
.58966 
.59013 


.72921 
.73010 
.73099 


.80799 
.80765 
.80730 


1.2376 
1.2382 
1.2387 


6 

8 
10 


.60321 
.60367 
.60413 


.75629 
.75721 
.75812 


.79758 
.79723 
.79688 


1.2538 
1.2543 
1.2549 


12 

14 
16 


.59060 
.59107 
.59154 


.73189 
.73278 
.73367 


.80696 
.80662 
.80627 


1.2392 
1.2397 
1.2403 


12 
14 
16 


.60460 
.60506 
.60552 


.75904 
.75996 
.7o087 


.79653 
'79618 
.79583 


1.2554 
1.2560 
1.2565 


18 
20 
22 


.59201 
.59248 
.59295 


.73157 

.73547 
.73636 


.80593 
.80558 
.80524 


1.2408 
1.2413 
1.2419 


18 
20 
22 


.60599 
.60645 
.60691 


.76179 
.76271 
.7b363 


.79547 
.79512 
.79477 


1.2571 

1.2577 
1.2582 


24 
26 
28 


.59342 
.59389 
.594,55 


.73726 
.73816 
.73906 


.80489 
.80455 
.80420 


1.2424 
1.2429 
1.2435 ! 


24 
26 

28 


.60738 
.60784 
.60830 


.76456 
.76548 
.76640 


.79441 
.79406 
.79371 


1.2588 
1.2593 
1.2599 


30 
32 
34 


.59482 
.59529 
.59576 


.73996 
.74086 
.74176 


.80386 
.80351 
.80316 


1.2440 
1.2445 
1.2451 


30 
32 
34 


.60876 
.60922 
.60968 


.76733 
.76825 
.76918 


.79335 
.79300 
.79264 


1.2605 
1.2610 
1.2616 


36 

38 
40 


.59622 
. 9669 
.59716 


.74266 
.74357 
.74447 


.80282 
.80247 
.8U212 


1.2456 
1.2461 
1.2466 


36 

38 
40 


.61014 
.61060 
.61107 


.77010 
.77103 
.77196 


.79229 
.79193 
.79158 


1.2622 
1.2627 
1.2633 


42 

44 
46 


.59762 
.59809 
.59856 


.74538 
.74628 
.74719 


.80177 
.80143 
.80108 


1.2472 

1.2478 
1.2483 


42 
44 
46 


.61153 
.61198 
.61245 


.77289 
.77382 

.77475 


.79122 
.79087 
.79051 


1.2639 
1.2644 
1.2650 


48 
50 
52 


.59902 
.59919 
.69995 


.74809 
.7491-0 
.74991 


.80073 
.80038 
.80003 


1.2488 
1.2494 

1.2499 


48 
50 
52 


.61291 
.61337 
.61382 


.77568 
.77661 
.77754 


.79015 
.78980 
.78944 


1.2656 
1.2661 
1.2667 


54 
56 

58 


.60042 
.60088 
.60135 


.75082 
.75173 
.75264 


.79968 
.79933 
.79898 


1.2505 
1.2510 

1.2516 


54 
56 
58 


.61428 
.61474 
.61520 


.77848 
.77941 

.78035 


.78908 
.78872 
.78837 


1.2673 

1.2679 
1.2684 



84 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 



NATURAL SINKS, TANGENTS, COSINES 
AND SECANTS. 







38 Deg. 








39 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.61566 
.61612 
.61658 


.78128 
.78222 
.78316 


.78801 
.78765 
.78729 


1.2690 
1.2696 
1.2702 



2 

4 


.62932 

.62977 
.63022 


.80978 
•81075 
.81171 


.77714 
.77678 
.77641 


1.2867 
1.2874 
1.2880 


6 

8 

10 


.61703 
.61749 
,61795 


.78410 
.78504 
.78598 


.78693 
.78657 
.78621 


1.2707 
1.2713 
1.2719 


6 

8 

10 


.63067 
.63113 
.63158 


.81268 
.81364 
.81461 


.77604 
.77568 
.77531 


1.2886 
1.2892 
1.2&8 


12 
14 
16 


.61841 
.61886 
.61932 


.78692 
.78786 
.78881 


.78586 
.78550 
.78514 


1.2725 
1.2731 
1.2737 


12 
14 
16 


.63203 
.63248 
.63293 


.81558 
.81655 
.81752 


.77494 
.77458 
.77421 


1.2904 
1.2910 
1.2916 


18 
20 
22 


.61978 
.62023 
.62069 


.78975 
.79070 
.79164 


.78477 
.78441 

.78405 


1.2742 
1.2748 
1.2754 


18 
20 
22 


.63338 
.63383 
.63428 


.81849 
.81946 
.82043 


.77384 
.77347 
.77310 


1.2922 
1.2929 
1.2935 


24 
26 
28 


.62115 
.62160 
.62206 


.79259 
.79354 
.79448 


.78369 
.78333 
.78297 


1.2760 
1.2766 

1.2772 


24 
26 
28 


.63473 
.63518 
.63563 


.82141 
.82238 
.82336 


.77273 
.77236 
.77199 


1.2941 

1.2947 
1.2953 


30 
32 
34 


.62251 
.62297 
.62342 


.79543 
.79638 
.79734 


.78261 
.78224 
.78188 


1.2778 
1.2784 
1.2790 


30 
32 

34 


.63608 
.63653 
.63697 


.82433 
.82531 
.82629 


.77162 
.77125 

.77088 


1.2960 
1.296(5 
1.2972 


36 

38 
40 


.62388 
.62433 
.62479 


.79829 
.79924 
.80019 


.78152 
.78115 
.78079 


1.2796 
1.2801 
1.2807 


36 

38 
40 


.63742 
.63787 
.63832 


.82727 
.82825 
.82923 


.77051 
.77014 

.76977 


1.2978 
1.2985 
1.2991 


42 
44 
46 


.62524 
.62570 
.62615 


.80115 
.80210 
.80306 


.78043 
.78006 
.77970 


1.2813 
1.2819 

1.2825 


42 
44 
46 


.63877 
.63921 
.63966 


.83021 
.83120 
.83218 


.76940 

.76903 
.76865 


1.2997 
1.3003 
1.3010 


48 
50 
52 


.62660 
.62706 
.62751 


.80402 

.80498 
.80594 


.77934 

.77897 
.77861 


1.2831 
1.2837 
1.2843 


48 
50 
52 


.64011 

.64055 
.64100 


.83317 
.83415 
.83514 


.76828 
.76791 
.76754 


1.3016 
1.3022 
1.3029 


54 
56 
58 


.62796 
.62841 
.62887 


.80690 
.80786 
.80882 


.77824 
.77788 
.77751 


1.2849 
1.2855 
1.2861 


54 
56 
58 


.64145 
.64190 
.64234 


.83613 

.83712 
.83811 


.76716 
.76679 
.76642 


1.3035 
1.3041 
1.3048 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







40 Deg. 








41 Deg. 




MIS 


SINE. 


TANG. 


COSI.NE. 


SEC- j 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.64279 
.64323 
.64368 


.83910 
.84009 
.84108 


.76604 
.76567 
.76529 


1.3054 
1.3060 
1.3067 




2 
4 


.65606 
.65650 
.6o694 


.86928 
.87u31 
,87133 


.75471 
.75433 
.75394 


1.3250 

13257 
1.3263 


6 

8 
10 


.64412 
.64457 
.64501 


.84208 
.84307 
.84407 


.76492 
.76455 
.76417 


1.3073 
1.3OS0 
1.3086 


6 

8 
10 


.65737 
.65781 
.65825 


.87235 
.87338 
.87440 


.75356 
.75318 
.75280 


1.3270 
1.3277 

1.3284 


12 
14 
16 


.64546 
.64690 
.64634 


.84506 
.84606 
.84706 


.76379 
.76342 
.76304 


1.3092 
1.3099 
1.3105 


12 
14 
16 


.65869 
.65913 
.65956 


.87543 
.87646 
.87749 


.75241 
'75203 
.75165 


1.3290 
1.3297 
1.3304 


18 
20 
22 


.64679 
.64723 
.61768 


.84806 
.84906 
.85006 


.76267 
.76229 
.76191 


1.3112 
1.3118 
1.3125 


18 
20 
22 

24 
26 
28 


.66000 
.66044 
.66087 


.87852 
.87955 
.88058 


.75126 
.75088 
.75049 


1.3311 
1.3318 
1.3324 


24 
26 

28 


.64812 
,61856 
.61900 


.85106 
.85207 
.85307 


.76154 
.76116 
.76078 


1.3131 
1.3138 
1.3144 


.66131 
.66175 
.66218 


.88162 
.88265 
.88369 


.75011 

.74972 
.74934 


1.3331 
1.3338 
1.3345 


30 
32 
34 


.64945 
.64989 
.65033 


.85408 
.85509 
.85609 


.76040 
•760U3 
.75965 


1.3151 
1.3157 
1.3164 


30 
32 
34 


.66262 

.66305 
.66348 


.88472 

.88576 
.88680 


.74895 
.74857 

.74818 


1.3352 
1.3359 
1.3366 


36 

38 
40 


.65077 
.65121 
.65166 


.85710 
.86811 

.85912 


.75927 
.75889 
.75851 


1.3170 
1.3177 
1.3184 


36 

38 
40 


.66392 
.66436 
.66479 


.88784 
.88888 
.88992 


.74780 
.74741 
.74702 


1.3372 
1.3379 
1.3386 


42 
44 
46 


.65210 
.65254 
.65298 


.86013 
.86115 
.86216 


.75813 
.75775 
.75737 


1.3190 
1.3197 
1.3202 


42 
44 
46 


.66523 
.66566 
.66610 


.89097 
.89201 
.89305 


.74664 
.74625 
.74586 


1.3393 
1.3400 
1.3407 


48 
50 
52 


.65342 
.65386 
.65430 


.86317 
.86419 
.86521 


.75699 
.75661 
.75623 


1.3210 
1.3217 

1.3223 


48 
50 
52 


.66653 
.66696 
.66740 


.89410 
.89515 
.89620 


.74547 
.74509 
.74470 


1.3414 
1.3421 
1.3428 


54 
56 

58 


.65474 
.65518 
.65562 


.86623 
.86724 
.86826 


.75585 
.75547 
.75509 


1.3230 
1.3237 
1.3243 


54 
56 

58 


.66783 
.66826 
.66870 


.89725 
.89830 
.89935 


.74431 
.74392 
•74353 


1.3435 
1.3442 
1.3449 



86 



THE MACHINIST AND TOOI^ MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







42 DEG. 


i 






43 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SIMS. 


TANG. 


COSINE. 


SEC. 



2 

4 


.66913 
.66956 
.66999 


.90040 
.90146 
.90251 


.74314 
.74275 
.74236 


1.3456 
1.3463 
1.3470 



2 

4 


.68200 
.68242 
.68285 


.93251 
.93360 
.93469 


.73135 
.73096 
/J3056 


1.3673 
1.368 L 

1.3688 


6 

8 
10 


.67043 

.67086 
.67129 


.90357 
.90462 
.90568 


.74197 
.74158 
.74119 


1.3477 
1.3485 
1.3492 


6 

8 

10 


.68327 
.68370 
.68412 


.93578 
.93687 
.93797 


.73016 
.72976 
.72937 


1.3695 
1.3703 
1.3710 


12 
14 
16 


.67172 
.67215 
.67258 


.90674 
.90780 
.90887 


.74080 
.74041 

.74002 


1.3499 
1.3506 
1.3513 


12 
14 
16 


.68455 
.68497 
.68539 


.93906 
.94016 
.94125 


.72897 
.72857 
.72817 


1.3718 
1.3725 
1.3733 


18 
20 
22 


.67301 
.67344 

.67387 


.90993 
.91099 
.91206 


.73963 
.73924 

.73885 


1.3520 
1.3527 
1.3534 


18 
20 
22 


.68582 
.68624 
.68666 


.94235 
.94345 
.94455 


.72777 
.72737 
.72697 


1.3740 
1.3748 
1.3756 


24 
26 
28 


.67430 
.67473 
.67516 


.91312 
.91419 
.91526 


.73845 
.73806 
.73767 


1.3542 
1.3549 
1.3556 


24 
26 
28 


.68709 
.68751 
.68793 


.94565 
.94675 
.94786 


.72657 
.72617 
.72577 


1.3763 
1.3771 
1.3778 


30 
32 
34 


.67559 
.67602 
.67645 


.91633 
.91740 
.91847 


.73728 
.73688 
.73649 


1.3563 
1.3571 
1.3578 


30 
32 
34 


.68835 
.68877 
.68920 


.94896 
.95007 
.95118 


.72537 
.72497 
.72457 


1.3786 
1.3794 
1.3801 


36 

38 
40 


.67687 
.67730 
.67773 


.91955 
.92062 
.92169 


.73609 
.73570 
.73531 


1.3585 
1.3592 
1.3600 


36 

38 
40 


.68962 
.69004 
.69046 


.95228 
.95339 
.95451 


.72417 

.72377 
.72337 


1.3809 
1.3816 
1.3824 


42 
44 
46 


.67816 
.67859 
.67901 


.92277 
.92385 
.92493 


.73491 
.73452 
.73412 


1.3607 
1.3614 
1.3622 


42 
44 
46 


.69088 
.69130 
.69172 


.95562 
.9.673 

.95785 


.72297 
.72256 
.72216 


1.3832 
1.3839 
1.3847 


48 
50 
52 


.67944 

.67987 
.68029 


.92601 
.92709 
.92817 


.73373 
.73333 
.73294 


1.3629 
1.3636 
1.3644 


48 
50 
52 


.69214 
.69256 
.69298 


.95896 
.96008 
.96120 


.72176 
.72136 
.72095 


1.3855 
1.3863 
1.3870 


54 
56 
58 


.68072 
.68114 
.68157 


.92926 
.93034 
.93143 


.73254 
.73215 
.73175 


1.3651 
1.3658 
1.3666 


54 
56 
58 


.69340 
.69382 
.69424 


.96232 
.96344 
.96456 


.72055 
.72015 
.71974 


1.3878 
1.3886 
1.3894 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



87 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







44 Deg. 








45 Deg. 




MIN 


SINK. 


TANG. 


COSINE. 


SEC 


MIN 


bINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.69466 
.69507 
.69549 


.96569 
.96681 
.96794 


.71934 
.71893 
.71853 


1.3902 
1.3909 
1.3917 




2 

4 


.70711 

.70, o2 
.70793 


1. 

1,0012 
1.0023 


.70711 

.70669 
.70628 


1.4142 
1.4150 
1.4159 


6 

8 
10 


.69591 
.69633 
.69675 


.96906 
.97019 
.97132 


.71812 
.71772 
.71732 


1.3925 
1.3933 
1.3941 


6 

8 
10 


.70834 
.70875 
.70916 


1.0035 
1.0047 
1.0058 


.70587 
.70546 
.70505 


1.4167 

1.41.5 
1.4183 


12 
14 
16 


.69716 
.69758 
.69800 


.97246 
.97359 
.97472 


.71691 
.71650 
.71610 


1.3949 
1.3957 
1.3964 


12 
14 
16 


.70957 
.70998 
.71039 


1.0070 
1.0082 
1.0093 


.70463 
•70422 
.70381 


1.4192 
1.4200 

1.4208 


18 
20 
22 


.69841 
.69883 
.69925 


.97586 
.97699 
.97813 


.71569 
.71528 
.71488 


1.3972 

1.3980 
1.3988 


18 
20 
22 


.71080 
.71121 
.71162 


1.0105 
1.0117 
1.0129 


.70339 
.70298 
.70257 


1.4217 
1.4225 
1.4233 


24 
26 
28 


.69966 
.70008 
.70049 


.97927 
.9804 L 
.98155 


.71447 
.71406 
.71366 


1.3996 
1.4004 
1.4012 


24 
26 
28 


.71202 
.71243 
.71284 


1.0141 
1.0152 
1.0164 


.70215 
.70174 
.70132 


1.4242 

1.4250 
1.4259 


30 
32 
34 


.70091 
.70132 
.70174 


.98270 
.98384 
.98499 


.71325 

.71284 
.71243 


1.4020 
1.4028 
1.4036 


30 
32 
34 


.71325 

.71366 
.71406 


1.0176 

1.0188 
1.0200 


.70091 
.70049 
.70008 


1.4267 
1.4276 
1.4284 


36 

38 
40 


.70215 
.70257 
.70298 


.98613 

.98728 
.98843 


.71202 
.71162 
.71121 


1.4044 
1.4052 
1.4060 


36 
38 
40 


.71447 
.71488 
.71528 


1.0212 
1.0223 
1.0235 


.69666 
.69925 
.69883 


1.4292 
1.4301 
1.4310 


42 
44 
46 


.70339 
.70381 
.70422 


.98958 
.99073 
.99 J 89 


.71080 
.71039 
.70998 


1.4069 
1.4077 
1.4085 


42 
44 
46 


.71569 
.71610 
.71650 


1.0247 
1.0259 
1.0271 


.69841 
.69800 
.69758 


1.4318 
1.4327 
1.43.5 


48 
50 
52 


.70463 
.70505 
.705,16 


.99304 
.99420 
.99535 


.70957 
.70916 
.70875 


1.4093 
L.4101 
1.4109 


48 
50 
52 


.71691 
.71732 
.71772 


1.0283 
1.0295 
1.0307 


.69716 
.69675 
.69633 


1.4344 
1.4352 
1.4361 


54 
56 
58 


.70587 
.70628 
.70669 


.99651 
.99767 
.99884 


.70834 
.70793 
.70752 


1.4117 
1.4126 
1.4134 


54 
56 
58 


.71812 
.71853 
.71893 


1.0319 
1.0331 
1.0343 


.69591 
.69549 
.69507 


1.4370 
1.4378 
1.4387 



88 



THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. 



NATURAL SINKS, TANGENTS, COSINES 
AND SECANTS. 







46 Deg. 


i 






47 DEG. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SIKE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.71934 
.71974 
.72015 


1.0355 
1.0367 
1.0379 


.69466 
.69424 
.69382 


1.4395 
1.4414 
1.4413 



2 

4 


.73135 
.73175. 

.73215 . 


1.0724 
1.0736 
1.0748 


.68200 
.68157 
.68115 


1.4663 
1.4672 
1.4681 


6 

8 
10 


.72055 
.72095 
72136 


1.0391 
1.0403 
1.0416 


.69310 
.69298 
.69256 


1.4422 
1.4430 
1.4439 


6 

8 

10 

12 

14 
16 

18 
20 
22 


.73254 
.73294 
.73333 


1.0761 
1.0774 
1.0786 


.68072 
.68029 
.67987 


1.4690 
1.4699 

l.47oy 


12 

14 
16 


.72176 

.72216 
.72256 


1.0428 
1.0440 
1.0452 


.69214 
.69172 
.69130 


1.4448 
1.4457 
1.4465 


.73373 
.73412 
.73462 


1.0799 
1.0812 
1.0824 


.67944 
.67901 
.67859 


1.4718 

1.4727 
1.4736 


18 
20 
22 


.72297 
.72337 

.72377 


1.0461 
1.0476 
1.048y 


.69088 
.69046 
.69004 


1.4474 
1.4483 
1.4492 


.73491 
.73 )31 
.73570 


1.0837 
1.0849 
1.0862 


.67816 
.67773 
.67730 


1.4745 
1.4755 
1.4764 


24 
26 
28 


.72417 

.72457 
.72497 


1.0501 
1.0513 
1.0525 


.68962 
.68920 
.68878 


1.4501 
1.4510 
1.4518 


24 
26 
28 


.73610 
.73649 
.73688 


1.0875 

1.0887 
1.0900 


.67687 
.67645 
.67602 


1.4774 

1.4783 
1.4792 


30 
32 
34 


.72537 
.72577 
.72617 


1.0538 
1.0550 
1.0562 


.68835 
.68793 
.68751 


1.4527 
1.4536 
1.4545 


30 
32 

34 


.73728 
.73767 
.73806 


1.0913 
1.0^26 
1.0938 


.67559 
.67516 
.67473 


1.4802 
1.4811 
1.4821 


36 
38 
40 


.72657 
.72697 
.72737 


1.0575 
1.0587 
1.0599 


.68709 
.68666 
.68624 


1.4554 
1.4563 
1.4572 


36 

38 
40 


.73845 

.73885 
.73924 


1.0951 
1.0964 
1.0977 


.67430 
.67387 
.67344 


1.4830 
1.4839 
1.4849 


42 
44 
46 


.72777 
.72817 
.72857 


1.0612 
1.0624 
1.0636 


.68582 
.68539 
.68197 


1.4581 
1.4590 
1.4599 


42 
44 
46 


.73963 

.74002 
.74041 


1.0990 
1.1003 
1.1015 


.67301 

.67258 
.67215 


1.4858 
1.4868 

1.4877 


48 
50 
52 


.72897 
.72937 
.72976 


1.0649 
1,0661 
1.0674 


.68455 
.68412 
.68370 


1.4608 
1.4617 
1.4626 


48 
50 
52 


.74080 
.74119 
.74158 


1.1028 
1.1041 
1.1054 


.67172 
.67129 
.67086 


1.4887 
1.4897 
1.4906 


54 
56 
(>8 


.73016 

.73056 
.73096 


1.0686 
1.0699 
1.0711 


.68327 
.68285 
.68242 


1.4635 
1.4644 
1.4654 


54 
56 

58 


.74197 
.74236 
.74275 


1.1067 
1.1080 
1.1093 


.67043 

.66999 
.66956 


1.4916 
1.4925 
1.4935 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







48 DEG. 


I 






49 DEG. 




MIS 


SINE. 


TANG. 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.74314 
.74353 
.74392 


1.11061 
1.11191 
1.11321 


.66913 
.66870 
.66827 


1.4946 
1.4954 
1.4964 



2 

4 


.75471 

.75509 
.75547 


1.15037 
1.15172 
1.15308 


.65606 
.65562 
.65518 


1.5242 
1 5253 
1.5263 


6 

8 

10 


.74431 
.74470 
.74509 


1.11452 
1.11582 
1.11713 


.66783 
.66740 
.66697 


1.4974 
1.4983 
1.49^3 


6 

8 

10 


.75585 
.75623 
.75661 


1.15443 
1J5579 

1.15715 


.65474 
.65430 
.65386 


1.5273 
1.5283 
1.5294 


12 

14 
16 


.74548 
.74586 
.74625 


1.11844 
1.11975 
1.12106 


.66653 
.66610 
.66566 


1.5003 
1.5013 
1.5022 


12 
14 
16 


.75700 

.75738 
.75775 


1.15851 
1.15987 
1.16124 


.65342 
.65298 
.65254 


1.5304 
1.5314 
1.5325 


18 
20 
22 


.74664 
.74703 
.74741 


1.12238 
1.12369 
1.12501 


.66523 
.66480 
.66136 


1.5032 
1.5042 
1.5052 


18 
20 
22 


.75813 
.75851 
.75889 


1.16261 
1.16398 
1.16535 


.65210 
.65166 
.65122 


1.5335 
1.5345 
1.5356 


24 
26 
28 


.74780 
.74818 
.74857 


1.12633 
1.12765 
1.12897 


.66393 
.66349 
.66306 


1.5062 
1.5072 
1.5082 


24 
26 
28 


.75927 
.75965 
.76003 


1.16672 
1.16809 
1.16947 


.65077 
.65033 
.64989 


1.5366 
1.5377 

1.5387 


30 
32 
34 


.74896 
.74934 
.74973 


1.13029 
1.13162 
3.13295 


.66262 
.66218 
.66175 


1.5092 
1.5101 
1.5111 


30 
32 
34 


.76041 
.76078 
.76116 


1.17085 
1.17223 
1.17361 


.64945 
.61901 
.64856 


1.5398 
1.5408 
1.5419 


36 
38 
40 


.75011 
.75050 
.75088 


1.L3428 
1.13561 
1.13694 


.66131 
.66088 
.66044 


1.5121 
1.5131 
1.5141 


36 
38 
40 


.76154 
.76192 
.76229 


1.17500 
1.17638 
1.17776 


.64812 

.64768 
.64723 


1.5429 
1.5440 
1.5450 


42 
44 
46 


.75126 
.75165 
.75203 


1.13828 
1.13961 
1.14095 


.66000 
.65956 
.65913 


1.5151 
1.5161 
1.5171 J 


42 
44 
46 


.76267 
.76304 
.76342 


1.17916 

1.18055 
1.18194 


.64679 
.64635 
.64590 


1.5461 
1.5471 

1.5482 


48 
50 
52 


.75241 
.75280 
.75318 


1.14229 
1.14363 
1.14498 


.65869 
.65825 
.65781 


1.5182 ! 
1.5192 
1.5202 ; 


48 
50 
52 


.76380 
.76417 
.76455 


1.18334 
1.18474 
1.18614 


.64546 
.64501 
.64457 


1.5493 
1.5503 
1.5514 


54 
56 
08 


.75356 
.75395 
.75433 


1.14632 
1.14767 
1.14902 


.65738 
.65694 
.65650 


1.5212 ! 

1.5222 

1.5232 


54 
56 
5S 


.76492 
.76530 
.76567 


1.18754 
1.18894 
1.19035 


.64412 
.64368 
.64323 


1.5525 

1.5536 
1.5546 



90 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







50 Deg. 








51 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.76604 
.76642 
.76679 


1.19175 

1.19316 
1.19457 


.64279 
.64234 
.64 J 90 


1.5557 
1.5568 
1 .5579 



2 

4 


.77715 
.77751 

.77788 


1.23490 
1.23637 
1.23784 


.62932 
.62887 
.62842 


1.5890 
1.5901 
1.5913 


6 

8 

10 


.76717 
.76754 
76791 


1.19599 
1.19740 

1.19862 


.64145 
.64100 
.64056 


1.5590 
1.560U 
1.5611 


6 

8 

10 

12 
14 
16 


.77824 
.77861 
.77897 


1.23931 
1.24079 
1.24227 


.62796 
.62751 
.62706 


1.5924 
1.5936 
1.5947 


12 
14 
16 


.76828 
.76866 
.76903 


1.20024 
1.20166 
1.20308 


.64011 
.63y66 
.63922 


1.5622 
1.5633 
1.5644 


.77934 

.77970 
.78007 


1.24375 
1.24523 
1.24672 


.62660 
.62615 
.62570 


1.5959 
1.5971 

1.5982 


18 
20 
22 


.76940 
.76977 
.77014 


1.20451 
1.20593 
1.20736 


.63877 
.63832 
.63787 


1.5655 
1.5666 
1 .5677 


18 
20 
22 


.78043 
.78079 
.78116 


1.24820 
1.24969 
1.25118 


.62524 
.62479 
.62433 


1.5994 
1.6005 
1.6017 


24 
26 
28 


.77051 

.77088 
.77125 


1.20879 
1.21023 

1.21166 


.63742 
.63698 
.63653 


1.5688 
1.5699 
1.5710 


24 
26 
28 


.78152 
.78188 
.78225 


1.25268 
1.25417 
1.25567 


.62388 
.62342 
.62297 


1.6029 
1.6040 
1.6052 


30 
32 
34 


.77162 
.77199 
.77236 


1.21310 
1.21454 
1.21598 


.63608 
.63563 
.63518 


1.5721 

1.5732 
1.5743 


30 
32 
34 


.78261 
.78296 
.78333 


1.25717 
1.25867 
1.26018 


.62251 
.62206 
.62160 


1.6064 
1.6077 
1.6088 


36 
38 
40 


.77273 
.77310 

.77347 


1.21742 

1.21885 
1.22031 


.63473 

.63428 
.63383 


1.5755 
1.5766 
1.5777 


36 

38 
40 


.78369 
.78405 
.78442 


1.26169 
1.26319 
1.26471 


.62115 
.62069 
.6-024 


1.6099 
1.6111 
1.6123 


42 
44 
46 


.77384 
.77421 
.77458 


1.22176 
1.22321 
1.22467 


.63338 
.63293 
.63248 


1.5788 
1.5799 
1.5811 


42 
44 
46 


.78478 
.78514 
.78j50 


1.26622 
1.26775 
1.26925 


.61978 
.61932 

.61887 


1.6135 
1.6147 
1.6159 


48 
50 
52 


.77494 
.77531 

.77568 


1.22612 

1.22758 
1.22904 


.63203 
.63158 
.63113 


1.5822 
1.5833 
1.5845 


48 
50 
52 


.78586 
.78622 
.78658 


1.27077 
1.27230 
1.27382 


.61841 
.61795 
.61749 


1.6170 
1.6182 
1.6194 


54 
56 
58 


.77605 
.77641 
.77677 


1.23050 
1.23196 
1.23343 


.63068 
.63022 
.62977 


1.5856 

1.5867 
1.5879 


54 
56 
58 


.78694 
.78729 
.78765 


1.27535 
1.27688 
1.27841 


.61704 
.61658 
.61612 


1.6206 
1.6218 
1.6231 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



9L 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







52 Deg. 








53 Deg. 




MIN 


SINK. 


TANG. 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.78801 
.78837 
.78873 


1.27994 
1.28148 
1.28302 


.61566 
.61520 
.61474 


1.6243 
1.6255 
1.6267 



2 

4 


.79864 
.79899 
.79934 


1.32704 
1.32865 
1.33026 


.60182 
.60135 
.60089 


1.6618 
16629 
1.6642 


6 

8 
10 


.78908 
.78944 
.78980 


1.28456 
1.28610 
1.28764 


.61429 
.61383 
.61337 


1.6279 
1.6291 
1.63U3 


6 

8 

10 


.79:68 
.80003 
.80038 


1.33187 
1.3334y 
1.33511 


.60042 
.59994 
.59949 


1.6655 
1.66t>8 
1.6(581 


12 
14 
16 


.79016 
.790ol 
.79087 


1.28919 
1.29074 
1.29229 


.61291 
.61245 
.61199 


1.6316 

1.6328 
1.0340 


12 
14 
16 


.80073 
.80108 
.80143 


1.33673 

1.33835 
1.33998 


.59902 
.59856 
.59809 


1.6694 
1.6707 
1.6720 


18 
20 
22 


.79122 
.79158 
.79193 


1.29385 
1.29541 
1.29696 


.61153 
.61107 
.61061 


1.6352 
1.6365 
1.6577 


18 
20 
22 


.80178 
.80212 
.80247 


1.34160 
1.34323 
1,34487 


.59763 

.59716 
.59669 


1.6733 
1.6746 
1.6759 


24 
26 
28 


.79229 
.79264 
.79300 


1.29853 
1.30009 
1.30166 


.61015 
.60968 
.60922 


1.6389 
1.6402 
1.6414 


24 
26 
28 


.80282 
.80316 
.80351 


1.34650 
1.34814 
1.34978 


.59622 
.59576 
.59529 


1.6772 
1.6785 
1.6798 


30 
32 
34 


.79335 
.79371 
.79406 


1.30323 
1.30480 
1.30637 


.60876 
.60830 
.60784 


1.6427 
1.6439 
1.6452 


30 
32 

34 


.80386 
.80420 
.80455 


1.35142 
1.35307 
1.35472 


.59482 
.59436 
.59389 


1.6812 
1.6826 
1.683d 


36 

38 
40 


.79441 
.79477 
.79512 


1.30795 
1.30952 
1.31110 


.60738 
.60691 
.60645 


1.6464 
1.6477 
1.6489 


36 
38 
40 


.80489 
.80524 
.80558 


1.35637 

1.35802 
1.35968 


.59342 
.59295 
.59248 


1.6S51 
1.6865 
1.6878 


42 
44 
46 


.79547 
.79583 
.79618 


1.31269 
1.31427 
1.31586 


.60599 
.60553 
.60506 


1.6502 
1.6514 
1.6527 


42 
44 
46 


.80593 
.80627 
.80662 


1.36134 

1.36300 
1.36466 


.59201 
.59154 
.59108 


1.6891 
1.6905 
1.6918 


48 
50 
52 


.79653 
.79688 
.79723 


1.31745 
1.319U4 
1.32064 


.60460 
.60414 
.60367 


1.6540 
1.6552 
1.6565 


48 
50 
52 


.80696 
.80730 
.80765 


1.36633 
1.36800 
1.36967 


.59061 
.59014 
.58967 


1.6932 
1.6945 
1.6959 


54 
56 

58 


.79758 
.79793 
.79829 


1.32224 
1.32384 
1.32544 


.60321 
.60274 
.60228 


1.6578 
1.6591 
1.6603 


54 
56 
58 


.80799 
.80833 
.80867 


1.37134 
1.37302 
1.37470 


.58920 
.58873 
.58826 


1.6972 
1.6986 
1.6999 



92 



THE MACHINIST AND TOOL, MAKEE's INSTRUCTOR. 



NATURAL SINKS, TANGENTS, COSINES 
AND SECANTS. 







54 DEG. 








55 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.80902 
.80936 
.80970 


1.37638 
1.37807 
1.37976 


.58779 
.58731 
.58684 


1.7013 
1.7027 
1.7040 



2 

4 


.81915 
.81949 
.81982 


1.42815 
1.42992 
1.43169 


.57358 
.57310 
.57262 


1.7434 
1.7449 
1.7463 


6 

8 

10 


.81004 
.81038 
.81072 


1.38145 
1.38314 

1.38484 


.58637 
.58590 
.58543 


1.7054 

1.7068 
1.7081 


6 

8 

10 

12 

14 
16 

18 
20 
22 


.82015 
.82048 
.82082 


1.43347 
1.43525 

1.43703 


.57215 
.57167 
.57119 


1.7478 
1.7493 
1.7507 


12 
14 
16 


.81106 
.81140 
.81174 


1.38653 

1.38824 
1.36994 


.58496 
.58449 
.58401 


1.7095 
1.7109 
1.7123 


.82115 
.82148 
.82181 


1.43881 
1.44060 
1.44239 


.57071 

.57024 
.56976 


1.7522 
1.7537 
1.7551 


18 
20 
22 


.81208 
.81242 
.81^76 


1.39165 
1.39336 
1.39507 


.58354 
.58^07 
.58260 


1.7137 
1.7151 
1.7164 


.82214 
.82248 
.82281 


1.44418 
1.44598 
1.44778 


.56928 
.56880 
.56832 


1.7566 
1.7581 
1.7596 


24 
26 
28 


.81310 
.81344 
.81378 


1.39679 
1.39850 
1.40022 


.58212 
.58165 
.58118 


1.7178 
1.7192 
1.7206 


24 
26 

28 


.82314 
.82347 
.82380 


1.44958 
1.45139 
1.45320 


.56784 
.56736 
.56689 


1.7610 

1.7625 
1.7640 


30 
32 
34 


.81412 
.81445 
.81479 


1.40195 
1.40367 
] .40540 


.58070 
.58023 

.57976 


1.7220 

1.7234 
1.7249 


30 
32 
34 


.82413 
.82446 
.82478 


1.45501 
1.45682 
1.45864 


.56641 
.56593 
.56545 


1.7655 

1.7670 
1.7685 


36 
38 
40 


.81513 
.81546 
.81580 


1.40714 
1.40887 
1.41061 


.57928 
.57881 
.57833 


1.7263 

1.7277 
1.7291 


36 

38 

40 


.82511 

.82544 
.82577 


1.46046 
1.4622y 
1.46411 


.56497 
.56449 
.56401 


1.7700 
1.7715 
1.7730 


42 

44 
46 


.81614 

.81647 
.81681 


1.41235 
1.41409 
1.41584 


.57786 
.57738 
.57691 


1.7305 
1.7319 
1.7334 


42 
44 
46 

48 
50 
52 


.82610 

.82643 
.82675 


1.46595 
1.46778 
1.46962 


.56353 
.56305 
,56256 


1.7745 
1.7760 

1.7775 


48 
50 
52 


.8171.4 
.81748 
.81782 


1.41759 
1,41934 
1.42110 


.57643 
.57596 
.57548 


1.7348 
1.7362 
1.7377 


.82708 
.82741 

.82773 


1.47146 
1.47330 
1.47514 


.56208 
.56160 
.56112 


1.7791 
1.7806 
1.7821 


54 
56 
58 


.81815 

.81848 
.81882 


1.42286 
1.42462 
1.42638 


.57501 
.57453 
.57405 


1.7391 
1.7405 
1.7420 


54 
56 
58 


.82806 
.82839 
.82871 


1.47699 
1.47885 
1.48070 


.56064 
.56016 
.55968 


1.7837 
1.7852 
1.7867 



THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR^ 



9S 





NATURAL SINES, TANGENTS 
AND SECANTS. 


, COSINES 




56 Deg. 


57 Deg. 


MIK 


SINE. 


TANG. COSINE. 


SEC. 


MIN 


SINE. 


TANG. COSINE. 


SEC. 



2 

4 


.32904 
.8293t 
.82969 


1.48256 
1.48442 
1.48629 


.55919 
.55871 
.55823 


1.7883 1 

1.7898 

1.7914 



2 

4 


.83867 
.83899 
.83930 


1.53986 
1,54183 
1.54379 


.54464 
.54415 
.54366 


1.8361 
18377 
1.8394 


6 

8 
10 


.83001 
.83034 
.83066 


1.48816 
1=49003 
1.49190 


.55775 
.55726 
.55678 


1.7929 
1.7945 
1.7960 


6 

8 
10 


.83962 
.83994 
.84025 


1.54576 
1.54774 
1.54972 


.54317 

.54269 
.54220 


1.8410 
1.8427 

1.8443 


12 
14 
16 


.83098 
.83131 
c83163 


1.49378 
1.49566 

1.49755 


.55630 
.55581 

.55533 


1.7976 
1.7992 
1.8007 


12 

14 
16 


.84057 
.84088 
.84120 


1.55170 

1.55368 
1.55567 


.54171 

.54122 
.54073 


1.8460 

1.8477 
1.8493 


18 
20 
22 


.83195 
.832z8 
.83260 


1.49944 
1.50133 
1.50322 


.55484 
.55436 
.55388 


1.8023 
1.8039 
1.8054 


18 
20 
22 


.84151 
.84182 
.84214 


1.55766 
1.55966 
1.56165 


.54024 
.53975 
.53926 


1.8510 
1.8527 
1.8544 


24 
26 

28 


.83292 
.83324 
.83356 


1.50512 
1.50702 
1.50893 


.55339 
.55291 
.55242 


1.8070 
1.8086 
1.8102 


24 
26 
28 


.84245 
.8427 7 
.84308 


1.56366 
1.56566 
1.56767 


.53877 
.53828 
.53779 


1.8561 

1.8578 
1.8595 


30 
32 
34 


.83389 
.83421 
.83453 


1.51084 
1.51275 

1.51466 


.55194 
.55145 

.55097 


1.8118 
1.8134 
1.8150 


30 
32 

34 


.84339 
.84370 
.84402 


1.56969 
1.57170 

1.57372 


.53730 
.53681 

.53632 


1.8611 

1.8629 
1.8646 


36 

38 
40 


.83485 
.83517 
.83^49 


1.51658 
1.51850 
1.52043 


.55048 
.54999 
.54951 


1.8166 
1.8182 
1.8198 


36 

38 
40 


.84433 
.84464 
.84495 


1.57575 
1.57778 
1.57981 


.53583 
.53534 
.53484 


1.8663 
1.8680 
1.8697 


42 

44 
40 


.83581 
.83633 
.83645 


1.52235 
1.52429 
1.52622 


.54902 

.54854 
.54805 


1.8214 ' 
1.8230 1 
1.8246 


42 
44 
46 


.84526 
.84557 
.8M5SS 


1,58184 
1.58388 
1.58593 


.53435 
.53386 
.53337 


1.8714 
1.8731 
1.8749 


48 
50 
52 


.83676 
.83708 
.83740 


1.52816 
1.53010 
1.53205 


.54756 
.54708 
.54659 


1.8263 
1.8279 
1.8295 


48 
50 
52 


.84619 
.84650 
.84681 


1.58797 
1.59002 
1.59208 


.53288 
.53238 
.53189 


1.8766 
1.8783 
1.6801 


54 
56 

58 


.83772 
.83804 
.83835 


1.53400 
1.53595 
1.53791 


.54610 
.54561 
.54513 


1.8311 
1.8328 
1.8344 


54 
56 
58 


.84712 
.84743 
.84774 


1.59414 
1.59620 
1.59826 


.53140 
.53091 
=53041 


1.8818 
1.8836 
1.8853 



94 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







58 Deg. 








59 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.84805 
.84836 
.84866 


1.60033 
1.60241 
1.60449 


.52992 
.52943 
.52893 


1.8871 
1.8888 
1.8906 



2 

4 


.85717 
.85747 
.85777 


1.66428 
1.66647 
1.66867 


.51504 
.51454 
.51404 


1.9416 
1.9435 
1.9454 


6 

8 

10 


.84897 
.84928 
.84959 


1.60657 
1.60865 
1.61074 


.52844 
.52794 
.52745 


1.8924 
1.8941 
1.8959 


6 

8 

10 


.85806 
.85836 
.85866 


1.67088 
1.67309 
1.67530 


.51354 
.51304 
.51254 


1.9473 
1.9491 
1.9510 


12 
14 
16 


.84989 
.8o020 
.850ol 


1.61283 
1.61493 
1.61703 


.52696 
.52646 
.52597 


1.8977 
1.8995 
1.9013 


12 
14 
16 


.85896 
.85926 
.85956 


1.67752 
1.67974 
1.68196 


.51204 
.51154 
.51104 


1.9530 
1.9549 
1.9568 


18 
20 
22 


.85081 
.85112 
.85142 


1.61914 
1.62125 
1.62336 


.52547 
.52498 
.52448 


1.9030 
1.9048 
1.9066 


18 
20 
22 


.85985 
.86015 
.86045 


1.68419 
1.68643 
1.68866 


.51054 
.51004 
.50954 


1.9587 
1.9606 
1.9625 


24 
26 
28 


.85173 
.85203 
.85254 


1.62548 
1.62760 
1.62972 


.52399 
.52349 
.52299 


1.9084 
1.9102 
1.9121 


24 
26 
28 


.86074 
.86104 
.86133 


1.69091 
1.6W316 
1.69541 


.50904 
.50854 
.50804 


1.9645 
1.9664 
1.9683 


30 
32 
34 


.85264 
.85294 
.85325 


1.63185 
1.63398 
1.63612 


.52250 
.52200 
.52151 


1.9139 
1.9157 
1.9175 


30 
32 
34 


.86163 
.86192 
.86222 


1.69766 
1.69992 
1.70219 


.50754 
.50704 
.50654 


1.9703 
1.9722 
1.9742 


36 

38 
40 


.85355 
.85385 
.85416 


1.63826 
1.64041 
1.64256 


.52101 
.52051 
.52002 


1.9193 
1.9212 
1.9230 


36 

38 
40 


.86251 
.86281 
.86310 


1.70446 
1.70673 
1.70901 


.50603 
.50553 
.50503 


1.9761 
1.9781 
1.9801 


42 
44 
46 


.85446 

.8S476 
.85506 


1.64471 
1.64687 
1.64903 


.51952 
.51902 
.51852 


1.9248 
1.9267 
1.9285 


42 
44 
46 


.86340 
.86369 
.86398 


1.71129 
1.71358 

1.71588 


.50453 
.50403 
.50352 


1.9820 
1.9840 
1.9860 


48 
50 
52 


.85536 
.85567 
.85597 


1.65120 
1.65337 
1.65554 


.51803 
.51753 
.51703 


1.9304 
1.9322 
1.9341 


48 
50 
52 


.86427 
.86457 
.86186 


1.71817 
1.72047 

1.72278 


.50302 
.50252 
.50201 


1.9880 
1.9900 
1.9920 


54 
56 
58 


.85627 
.85657 
.85487 


1.65772 
1.65990 
1.66209 


.51653 
.51604 
.51554 


1.9360 
1.9378 
1.9397 


54 
56 
58 


,86515 
.86544 
.86573 


1.72509 
1.72741 
1.72973 


.50151 
.50101 
.50050 


1.9940 

1.9960 
1.9980 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 

NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 



60 Deg. 


61 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 




2 

4 


.86603 
.86632 
.86661 


1.73205 
1.73438 
1.73671 


.5 

.49950 
.49899 


2. 

2.002 

2.004 



2 

4 


.87462 
.87490 
.87518 


1.80405 
1,80653 
1.80901 


.48481 
.48430 
.48379 


2.063 
2 065 
2.067 


6 

8 

10 


.86690 
.86719 
.86748 


1.73905 
1.74140 
1.74375 


.49849 
.49798 
.49748 


2.006 
2.008 
2.010 


6 

8 

10 


.87546 
.87575 
.87603 


1.81150 
1.81399 
1.81649 


.48328 

.48277 
.48226 


2.069 
2.071 
2.073 


12 
14 
16 


.86777 
.86805 
.86834 


1.74610 
1.74846 
1.75082 


.49697 
.49647 
.49596 


2.012 
2.014 
2.016 


12 
14 
16 


.87631 
.87659 
.87b87 


1.81899 
1.82150 
1.82402 


.48175 
.48124 
.48073 


2.076 
2.078 

2.080 


18 
20 
22 


.86863 
.86892 
.86921 


1.75319 
1.75556 
1.75794 


.49546 
.49495 
.49445 


2.018 
2.020 
2.022 


18 
20 
22 


.87715 
.87743 
.87770 


1.82654 
1.82906 
L 83159 


.48022 
.47971 
.47920 


2.082 
2.085 
2.087 


24 
26 
28 


.86949 
.86978 
.87007 


1.76032 
1.76271 
1.76510 


.49394 
.49344 
.49293 


2.024 
2.026 
2.029 


24 
26 
28 


.87798 
.87826 
.87854 


1.83413 

1.83667 
1.83922 


.47869 
.47818 
.47767 


2.089 
2.091 
2.093 


30 
32 
34 


.87036 

.87064 
.87093 


1.76749 
1.76990 
1.77230 


.49242 
.49192 
.49141 


2.031 
2.033 
2.035 


30 
32 
34 


.87882 
.87909 
.87937 


1.84177 
1.84433 
1.84689 


.47716 
.47665 
.47614 


2.096 
2.098 
2.100 


36 

38 
40 


.87121 
.87150 
.87178 


1.77471 
1.77713 
1.77955 


.49090 
.49040 
.48999 


2.037 
2.039 
2.041 


36 
38 
40 


.87965 
.87993 
.88020 


1.84946 
1.85204 
1.85462 


.47562 
.47511 
.47460 


2.102 
2.105 
2.107 


42 
44 
46 


.87207 
.87235 
.87264 


1.78198 
1.78441 

1.78685 


.48938 

.48888 
.48837 


2.043 
2.045 
2.047 


42 
44 
46 


.88048 
.88075 
.88103 


1.85720 
1.85979 
1.86239 


.47409 

.47358 
.47306 


2.109 
2.112 
2.114 


48 
50 
52 


.87292 
.87321 
.87349 


1.78929 
1.79174 
1.79419 


.48786 
.48735 
.48684 


2.050 
2.052 
2.054 


48 
50 
52 


.88130 

.88158 
.88185 


1.86499 
1.86760 
1.87021 


.47255 
.47204 
.47153 


2.116 

2.1)8 
2,121 


54 
56 

58 


.87377 
.87406 
.87434 


1.79665 
1.79911 
1.80158 


.48634 
.48583 
.48532 


2.056 
2.058 
2.060 


54 
56 

58 


.88213 
.88240 
.88267 


1.87283 
1.87546 
1.87809 


.47101 
.47050 
.46999 


2.123 
2.125 
2.128 



96 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







62 Deg. 








63 Deg. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.88295 
.88322 
.88349 


1.88073 

1.88337 
1.88602 


.46947 
.46896 
.46844 


2.130 
2.132 
2.135 



2 

4 


.89101 
.89127 
.89153 


1.96261 
1.96544 
1.96827 


.45399 
.45347 
.45^95 


2.203 
2.205 
2.208 


6 

8 

10 


.88377 
88404 
.88431 


1.88867 
1.89133 
1.89400 


.46793 
.46742 
.46690 


2.137 
2.139 
2.142 


6 

8 
10 


.89180 
.89206 
.89232 


1.97111 

1.97395 
1.97681 


.45243 
.45192 
.45140 


2.210 
2.213 
2.215 


12 
14 
16 


.88458 
.88485 
.88512 


1.89667 
1.89935 
1.90203 


.46639 
.46587 
.46536 


2.144 
2.146 
2.149 


12 
14 
16 


.89259 
.89285 
.89311 


1.97966 
1.98253 
1.98540 


.45088 
.45036 
.44984 


2.218 
2.220 
2.223 


18 
20 
22 


.88539 
.88566 
.88593 


1.90472 
1.90741 
1.91012 


.46484 
.46433 
.46381 


2.151 
2.154 
2.156 


18 
20 
22 


.89337 
.89363 
.89389 


1.98828 
1.99116 
1.99406 


.44932 
.44880 
.44828 


2.226 
2.228 
2.231 


24 
26 
28 


.88620 
.88647 
.88674 


1.91282 
1.91554 
1.91826 


.46330 
.46278 
.46226 


2.158 
2.161 
2.163 


24 
26 
28 


.89415 
.89441 
.89467 


1.99695 
1.99986 
2.00277 


.44776 
.44724 
.44672 


2.233 
2.236 
2.238 


30 
32 
34 


.88701 
.88728 
.88755 


1.92098 
1.92371 
1.92645 


.46175 
.46123 
.46072 


2.166 
2.168 
2.170 


30 
32 
34 


.89493 
.89519 
.89545 


2.00569 
2.00862 
2.01155 


.44620 
.44568 
.44516 


2.241 

2.244 
2.246 


36 

38 
40 


.88782 
.88808 
.88835 


1.92920 
1.93195 
1.93470 


.46020 
.45968 
.45917 


2.173 

2.175 
2.178 


36 

38 
40 


.89571 
.89597 
.89623 


2.01449 
2.01743 
2.02039 


.44464 
.44411 
.44359 


2.249 
2.252 
2.254 


42 
44 
46 


.88862 
.88888 
.88915 


1.93746 
1.94023 
1.94301 


.45865 
.45813 
.45762 


2.180 
2.183 

2.185 


42 
44 
46 


.89649 
.89674 
.89700 


2.02335 
2.02631 
2.02929 


.44307 
.44255 
.44203 


2.257 
2.260 
2.262 


48 
50 
52 


.88942 
.88968 
.88995 


1.94579 
1.94858 
1.95137 


.45710 
.45658 
.45606 


2.188 
2.190 
2.193 


48 
50 
52 


.89726 
.89752 
.89777 


2.03227 
2.03526 
2.03825 


.44151 
.44098 
.44046 


2.265 
2.268 

2.27$ 


54 
56 

58 


.89021 
.89048 
.89074 


1.95417 
1.95698 
1.95979 


.45554 
.45503 
.45451 


2.195 

2.198 
2.200 


54 
56 

58 


.89803 
.89828 
.89854 


2.01125 

2.04426 
2.04728 


.43994 
.43942 
.43889 


2.273 
2.2,6 
2.2^8 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



97 



NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 



64 DEG. 


65 Deg. 


MIX 


SINE. 


TAXG- 


COSINE. 


SEC- 


MIX 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.89879 
.89905 
.89930 


2.0503 
2.0533 
2.0564 


.43837 
.43785 
.43733 


2.281 
2.284 

2.287 



2 

4 


.90631 
.90655 
.90680 


.21445 
.21478 
.21510 


.42262 

.42209 
.42156 


2.366 
2.369 
2.372 


6 

8 

10 


.89956 
.899*1 
.90007 


2.0594 
2.0625 
2.0655 


.43680 
.43628 
.43575 


2 289 
2.292 
2.295 


6 

8 

10 


.90704 
.90729 
.90753 


.21543 
.21576 
.21609 


.42104 
.42051 

.41998 


2.375 
2.378 
2.381 


12 
14 
16 


.90032 
.90057 
.90082 


2.0686 
2.0717 
2.0747 


.43523 
.43471 
.43418 


2.297 
2.300 
2.303 


12 
14 
16 


.90778 
.90802 
.90826 


.21642 
.21675 
.21708 


.41945 
.41892 
.41840 


2.384 
2.387 
2.390 


18 
20 
22 


.90108 
.90133 
.90158 


2.0778 
2.0809 
2.0840 


.43366 
.43313 
.43261 


2.306 
2.309 : 
2.311 


18 
20 
22 


.90851 
.90875 
.90899 


.21742 
.21775 
.21808 


.41787 
.41734 
.41681 


2.393 
2.396 
2.399 


24 
26 
28 


.90183 
.90208 
.90233 


2.0872 
2.0903 
2.0934 


.43209 
.43156 
.43104 


2.314 
2.317 
2.320 


24 
26 
28 


.90924 
.90948 
.90972 


.21842 
.21875 
.21909 


.41628 
.41575 
.41522 


2.402 
2.405 
2.408 


30 
32 
34 


.90259 

.90281 
.90309 


2.0965 
2.0997 
2.1028 


.43051 
.42999 
.42946 


2.323 

2.326 
2.328 


30 
32 
34 


.90996 
.91020 
.91044 


.21943 
.21977 
.22011 


.41469 
.41416 
.41363 


2.411 
2.414 
2.417 


36 
38 
40 


.90334 
.90358 
.90383 


2.1060 
2.1092 
2.1123 


.42894 
.42841 
.42788 


2.331 
2.334 
2.337 


36 
38 
40 


.91068 
.91092 
.91116 


.22045 
.22079 
.22113 


.418x0 

.41257 
.41204 


2.421 

2.424 
2.427 


42 
44 
46 


.90408 
.90433 
.90458 


2.1155 
2.1187 
2.1219 


.42736 
.42683 
.42631 


2.340 
2.343 
2 346 


42 
44 
46 


.91140 
.91164 
.91188 


.22147 
.22182 
.22216 


.41151 

.41098 
.41045 


2.430 
2.433 
2.436 


48 
50 
52 


.90483 
.90507 
.90532 


2.1251 
2.1283 
2.1315 


.42578 
.42525 
.42473 


2.349 
2.351 
2.a54 


48 
50 
52 


.91212 
.91236 
.91260 


.22251 
.22286 
.22320 


.40992 
.40939 
.40886 


2.439 
2.443 
2.446 


54 
56 
58 


.90557 
.90582 
.90606 


2.1348 
2.1380 
2.1412 


.42420 
.42367 
.42315 


2.357 
2.360 
2.363 


54 
56 

58 


.91283 
.91307 
.91331 


.22355 
.22390 
.22425 


.40833 
.40780 
.40727 


2.449 
2.452 
2.455 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



68 Deg. 


67 DEG. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIH 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


,91355 
.91378 
.91402 


2.2460 
2.2495 
2.2531 


.40674 
.40621 
.40567 


2.458 
2.462 
2.465 



2 

4 


.92050 
.92073 
.92096 


2.3558 
2.3597 
2.3635 


.39073 
.39020 
.38966 


2.559 
2.563 
2.566 


6 

8 

10 


.91425 
.91449 
.91472 


2.2566 
2.2602 
2.2637 


.40514 
.40461 
.40408 


2.468 
2.471 
2.475 


6 

8 
10 


.92119 
.92141 
.92164 


2.3673 
2.3712 
2.3750 


.38912 
.38859 
.38805 


2.570 
2.573 
2.577 


12 
14 
16 


.91496 
.91519 
,91543 


2.2673 
2.2709 
2.2745 


.40355 
.40301 
.40248 


2.478 
2.481 
2.485 


12 
14 
16 


.92186 
.92209 
.92231 


2.3789 
2.3828 
2.3867 


.38752 
.38698 
.38644 


2.580 
2.584 
2.588 


18 
20 
22 


.91566 
.91590 
.91613 


2.2781 
2.2816 
2.2853 


.40195 
.40141 

.40088 


2.488 
2.491 
2.494 


18 
20 
22 


.92254 
.92276 
.92299 


2.3906 
2.3945 
2.3984 


.38591 
.38537 
.38483 


2.591 
2.595 
2.598 


24 
26 
28 


.91636 
.91660 
.91683 


2.2889 
2.2925 
2.2962 


.40035 

.39982 
.39928 


2.498 
2.501 
2.504 


24 
26 
28 


.92321 
.92343 
.92366 


2.4023 
2.4063 
2.4102 


.38430 
.38376 
.38322 


2.602 
2.606 
2.609 


30 
32 
34 


.91706 
.91729 
.91752 


2.2998 
2.3035 
2.3072 


.39875 
.39822 
.39768 


2.508 
2.511 
2.515 


30 
32 
34 


.92388 
.92410 
.92432 


2.4142 
2.4182 
2.4222 


.38268 
.38215 
.38161 


2.613 
2.617 
2.620 


36 
38 
40 


.91775 
.91799 
.91822 


2.3108 
2.3145 
2.3183 


.39715 
.39661 
.39608 


2.518 
2.521 
2.525 


36 
38 
40 


.92455 
.92477 
.92499 


2.4262 
2.4302 
2.4.342 


.38107 
.38053 
.37^99 


2.624 
2.628 
2.632 


42 
44 
46 


.91845 
.91868 
.91891 


2.3219 
2.3257 
2.3294 


.39555 
.39501 
.39448 


2.528 
2.532 
2.535 


42 
44 
46 


.92521 
.92543 
.92565 


2.4382 
2.4423 
2.4464 


.37946 
.37892 
.37838 


2.635 
2.639 
2.643 


48 
50 
52 


.91914 
.91936 
.91959 


2.3332 
2.3369 
2.3407 


.39394 
.39341 
.39287 


2.538 
2.542 
2.545 


48 
50 
52 


.92587 
.92609 
.92631 


2.4504 
2.4545 
2.4586 


.37784 
.37730 
.37676 


2.647 
2.650 
2.654 


54 
56 

58 


.91982 
.92005 
.92028 


2.3445 

2.3482 
2.3520 


.39234 
.39180 
.39127 


2.549 
2.552 
2.556 


54 
56 
58 


.92653 
.92675 
.92697 


2.4627 
2.4668 
2.4709 


.37622 
.37569 
.37515 


2.658 
2.662 
2.666 



THE MACHINIST AND TOOD MAKER'S INSTRUCTOR. 



99 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







68 Dec. 








69 Deg. 




MIS 


SINK. 


TANG. 


COSINE. 


SEC 


MIN 


SINE. 


TANG. 


COSINE . 


SBC. 



2 

4 


.92718 
.92740 
.92762 


2.4751 
2.4792 
2.4834 


.37461 
.37407 
.37353 


2.669 
2.673 
2.677 



2 

4 


.93358 
.93379 
.93400 


2.6051 
2.6096 
2.6142 


.35837 
.35782 
.35728 


2.790 
2 795 

2.799 


6 

8 

10 


.92784 
.92805 
.92827 


2.4876 
2.4918 
2.4960 


.37299 
.37245 
.37191 


2.681 

2.685 
2.689 


6 

8 

10 


.93420 
.93441 
.93462 


2.6187 
2.6233 
2.6279 


.35674 
.35619 

.35565 


2.803 
2.807 
2.812 


12 
14 
16 


.92849 
.92870 
.92892 


2.5002 
2.5044 
2.5086 


.37137 

.37083 
.37029 


2.693 
2.697 
2.700 


12 
14 
16 


.93483 
.93503 
.93524 


2.6325 
2.6371 
2.6418 


.35511 
.35456 
.35402 


2.816 
2.820 
2.825 


18 
20 
22 


.92913 
.92935 
.92956 


2.5129 
2.5171 
2.5214 


.36975 
.36921 
.36867 


2.704 
2.708 
2.712 


18 
20 
22 


.93544 
.93565 
.93585 


2.6464 
2.6511 
2 6557 


.35347 
.35293 
.35239 


2.829 
2.833 
2.838 


24 
26 
28 


.92978 
.92999 
.93020 


2.5257 
2.5300 
2.5343 


.36812 
.36758 
.36704 


2.716 
2.720 
2.724 


24 
26 
28 


.93606 
.93626 
.93647 


2.6604 
2.6651 
2.6699 


.35184 
.35130 
.35075 


2.842 
2.847 
2.851 


30 
32 
34 


.93042 
.93063 
.93084 


2.5386 
2.5430 
2.5473 


.36650 
.36596 
.36542 


2.728 
2.732 
2.737 


30 
32 
34 


.93667 
.93688 
.93708 


2.6746 
2.6794 
2.6841 


.35021 
.34966 
.34912 


2.855 
2.860 
2,864 


36 

38 
40 


.93106 
.93127 
.93148 


2.5517 
2.5561 
2.5604 


.36488 
.36434 
.36379 


2.741 
2.745 
2.749 


36 
38 
40 


.93728 
.93748 
.93769 


2.6889 
2.6937 
2.6985 


.34857 
.34803 
.34748 


2.869 
2.873 

2.878 


42 
44 
46 


.93169 
.93190 
.93211 


2.5649 
2.5693 
2.5737 


.36325 
.36271 
.36217 


2.753 
2.757 
2.761 


42 
44 
46 


.93789 
.93809 
.93829 


2.7033 
2.7082 
2.7130 


.34694 
.34639 
.34584 


2.882 
2.887 
2.891 


48 
50 
52 


.93232 
.93253 

.93274 


2.5781 
2.5826 
2.5871 


.36162 
.36108 
.36054 


2.765 
2.769 
2.773 


48 
50 
52 


.93849 
.93869 
.93889 


2.7179 

2.7228 
2.7277 


.34530 
.34475 
.34421 


2.896 
2.900 
2.905 


54 
56 
58 


.93295 
.93316 
.93337 


2.5915 
2.5960 
2.6005 


.36000 
.35945 
.35891 


2.778 
2.782 
2.786 


54 
56 
58 


.93909 
.93929 
.93949 


2.7326 
2.7375 
2.7425 


3.4366 
3.4311 
3.4257 


2.910 
2.914 
2.919 



100 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







70 DEG. 








71 DEG. 




MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.93969 
.93989 
.94009 


2.7475 
2.7524 
2.7575 


.34202 
.34147 
.34093 


2.924 
2.928 
2.933 



2 

4 


.94552 
.94571 
.94590 


2.9042 
2.9097 
2.9152 


.32557 
.32502 
.32447 


3.071 
3.077 
3.082 


6 

8 

10 


M029- 

.94049 
.94068 


2.7625 
2.7675 

2.7725 


.34038 
.33983 
.33929 


2.938 
2.943 
2.947 


6 

8 

10 


.94609 
.94627 
.94646 


2.9207 
2.9263 
2.9319 


.32392 
.32337 

.32282 


3.087 
3.092 
3.098 


12 
14 
16 


.94088 
.94108 
.94127 


2.7776 

2.7827 
2.7878 


.33874 
.33819 
.33764 


2.952 
2.957 
2.962 


12 
14 
16 


.94665 
.94684 
.94702 


2.9375 
2 9431 

2.9487 


.32227 
.32171 
.32116 


3.103 
3 108 
3.1U 


18 
20 
22 


.94147 
.94167 
.94186 


2.7929 

2.7980 
2.8032 


33710 
.33655 
.33600 


2.966 
2.971 
2.976 


18 
20 
22 


.94721 
.94740 
.94758 


2.9544 
2.9600 
2.9657 


.32061 
.32006 
.31951 


3.119 
3.124 
3.130 


24 
26 
28 


.94206 
.94225 
.94245 


2.8083 
2.8135 
2.8187 


.33545 
.33490 
.33436 


2.981 
2.986 
2.991 


24 
26 
28 


.94777 
.94795 
.94814 


2.9714 

2.9772 
2.9829 


.31896 
.31841 
.31786 


3.135 
3.141 
3.146 


30 
32 
34 


.94264 
.94284 
.94303 


2.8239 
2.8291 
2.8344 


.33381 
.33326 
.33271 


2.996 

3. 

3.005 


30 
32 
34 


.94832 
.94851 
.94869 


2.9887 
2.9945 
3.0003 


.31730 
.31675 
.31620 


3.151 
3.157 
3.162 


36 
38 

40 


.94322 
.94342 
.94361 


2.8396 
2.8449 
2.8502 


.33216 
.33161 
.33106 


3 011 
3.016 
3.021 


36 
38 
40 


.94888 
.94906 
.94924 


3.0061 
3.0119 
3.0178 


.31565 
.31510 
.31454 


3.168 
3.174 
3.179 


42 
44 
46 


.94380 
.94399 
.94418 


2.8555 
2.8609 
2.8662 


.33051 

.32997 
.32942 


3.026 
3.031 
3.036 


42 
44 
46 


.94943 
.94961 
.94979 


3.0237 
3.0296 
3.0356 


.31399 
.31344 
.31289 


3.185 
3.190 
3.196 


48 
50 
52 


.94438 
.94457 
.94476 


2.8716 
2.8770 
2.8824 


.32887 
.32832 
.32777 


3.041 
3.046 
3.051 


48 
50 
52 


.94997 
.95015 
.95033 


3.0415 
3.0475 
3.0535 


.31233 
.31178 
.31123 


3.202 
3.207 
3.213 


54 
56 
58 


.94495 
.94514 
.94533 


2.8878 
2.8933 
2.8987 


.32722 
.32667 
.32612 


3.056 
3.061 
3.066 


54 
56 
58 


.95052 
.95070 
.95088 


3.0595 
3.0655 
3.0716 


.31068 
.31012 
.30957 


3.219 

3.224 
3.230 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



101 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



72 Deg. 


73 Deg. 


MIN 


SINE. 


TANG- 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC 



2 

4 


.95106 
.95124 
.95142 


3.0777 
3.0838 
3.0899 


.30902 
.30846 
.30791 


3.236 
3.242 
3.248 



2 

4 


.95630 
.95647 
.95664 


.32708 
.32777 
.32845 


.29237 
.29182 
.29126 


3.420 
3.427 
3.433 


6 

8 
10 


.95159 
.95177 
.95195 


3.0961 
3.1022 
3.1084 


.30736 
.30680 
.30625 


3.253 
3.259 
3.265 


6 

8 
10 


.95681 
.95698 
.95715 


.32914 

.32983 
.33052 


.29070 
.29015 
.28959 


3.440 
3.446 
3.453 


12 
14 
16 


.95213 
.9523 L 
.95248 


3.1146 
3.1209 
3.1271 


.30570 
.30514 
.30459 


3.271 
3.277 
3.283 


12 
14 
16 


.95732 
.95749 
.95766 


.33122 
.33191 
.33261 


.28903 
.28847 
.28792 


3.460 
3.466 
3.473 


18 
20 
22 


.95266 
.95284 
.95301 


3.1334 
3.1397 
3.1460 


.30403 
.30348 
.30292 


3.289 

3.295 
3.301 


18 
20 
22 


.95782 
.95799 
.95816 


.33332 
.33402 
.33473 


.28736 
.28680 
.28625 


3.480 
3.487 
3.493 


24 

26 

28 


.95319 
.95337 
.95354 


3.1524 
3.1588 
3.1652 


.30237 
.30182 
.30126 


3.307 
3.313 
3.319 


24 
26 
28 


.95832 
.95849 
.95865 


.33544 
.33616 
.33687 


.28569 
.28513 
.28457 


3.500 
3.507 
3.514 


30 
32 
34 


.95372 
.95389 
.95407 


3.1716 
3.1780 
3.1845 


.30071 
.30015 
.29960 


3.325 
3.332 
3.338 


30 
32 
34 


.95882 
.95898 
.95915 


.33759 
.33832 
.33904 


.28402 
.28346 
.28290 


3.521 
3.528 
3.535 


36 
38 
40 


.95424 
.95441 
.9M59 


3.1910 
3.1975 
3.2041 


.29904 
.29849 
.29793 


3.344 
3.350 
3.356 


36 
38 
40 


.95931 
.95948 
.95964 


.33977 
.34050 
.34124 


.28234 
.28178 
.28123 


3.542 
3.549 
3.556 


42 

44 
46 


.95476 
.95493 
.95511 


3.2106 
3.2172 
3.2238 


.29737 
.296*2 
.29626 


3.363 
3.369 
3 375 


42 
44 
46 


.95981 
.95997 
.96013 


.34197 
.34271 
.34346 


.28067 
.28011 
.27955 


3.563 
3.570 
3.577 


48 
50 
52 


.95528 
.95545 
.95562 


3.2305 
3.2371 
3.2438 


.29571 

.29515 
.29460 


3.382 
3.388 
3.394 


48 
50 
52 


.96029 
.96046 
.96062 


.34420 
.34495 
.34570 


.27899 
.27843 
.27787 


3.584 
3.591 
3.599 


54 
56 
58 


.95579 
.95596 
.95613 


3.2505 
3.2573 
3.2641 


.29404 
.29348 
.29293 


3.401 
3.407 
3.414 


54 
56 
58 


.96078 
.96094 
.96110 


.34646 
.3^722 
.34798 


.27731 
.27676 
.27620 


3.606 
3.613 
3.621 



102 



TKE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



74 DEG. 


75 Deg. 


MIN 


SINE. 


TANG. 


COSINE 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.96126 
.96142 
.96158 


3.4874 
8.4951 
3.5028 


.27564 
.27508 
.27452 


3.628 
3.635 
3.643 



2 

4 


.96593 
.96608 
.96623 


3.7320 
3.7407 
3.7495 


.25882 
.25826 
.25769 


3.864 

3.872 
3.880 


6 

8 

10 


.96174 
.96190 
.96206 


3.5105 
3.5183 
3.5261 


.27396 
.27340 

.27284 


3.650 
3.658 
3.665 


6 

8 
10 


.96638 
.96653 
.96667 


3.7583 
3.7671 
3.7759 


.25713 
.25657 
.25601 


3.899 
3.898 
3.906 


12 
14 
16 


.96222 
.96238 
,96253 


3.5339 
3.5418 
3.5497 


.27228 
.27172 
.27116 


3.673 
3.680 
3.688 


12 
14 
16 


.96682 
.96697 
.96712 


3.7848 
3.7938 
3.8028 


.25545 
.25488 
.25432 


3.915 
3.923 
3.932 


18 
20 
22 


.96269 
.96285 
.96301 


3.5576 
3.5656 
3.5736 


.27060 

.27004 
.26948 


3.695 
3.703 
3.711 


18 
20 
22 


.96727 
.96742 
.96756 


3.8118 
3.8208 
3.8299 


.25376 
.25320 
.25263 


3.941 
3.949 
3.958 


24 
26 
28 


.96316 
.96332 
.96347 


3.5816 
3.5897 
3.5977 


.26892 
.26836 
.26780 


3.719 
3.726 
3.734 


24 
26 
28 


.96771 
.96786 
.96800 


3.8391 

3.8482 
3.8574 


.25207 
.25151 
.25094 


8.967 
3.976 
3.985 


30 
32 
34 


.96363 
.96379 
.96394 


3.6059 
3.6140 
3.6222 


.26724 
.26668 
.26612 


3.742 
3.750 
3.758 


30 
32 
34 

36 

38 
40 


.96815 
.96829 
.96844 


3.8667 
3.8760 
3.8854 


.25038 
.24982 
.24925 


3.994 
4.003 
4.012 


36 
38 
40 


.96410 
.96425 
.96440 


3.6305 
3.6387 
3.6470 


.26556 
.26500 
.26443 


3.766 
3.774 
3.782 


.96858 
.96873 
.96887 


8.8947 
3.9042 
3.9136 


.24869 
.24813 
.24756 


4.021 
4.030 
4.039 


42 

44 
46 


.96456 
.96471 
.96486 


3.6554 
3.6638 
3.6722 


.26387 
.26331 
.26275 


3.790 
3.798 
3.806 


42 
44 
46 


.96902 
.96916 
.96930 


3.9232 
3.9327 
3.9423 


.24700 
.24644 
.24587 


4.049 
4.058 
4.067 


48 
50 
52 


.96502 
.96517 
.96532 


3.6806 
3.6891 
3.6976 


.26219 
.26163 
.26107 


3.814 
3.822 
3.830 


48 
50 
52 


.96945 
.96959 
.96973 


3.9520 
3.9616 
3.9714 


.24531 
.24474 
.24418 


4.076 
4.086 
4.095 


54 
56 
58 


.96547 
.96562 
.96578 


3.7062 
3.7148 
3.7234 


.26050 
.25994 
.25938 


3.839 
3.847 
3.855 


54 
56 
58 


.96987 
.97001 
.97015 


3.9812 
3.9910 
4.0009 


.24362 
.24305 
.24249 


4.105 
4.114 
4.124 



THE MACHINIST AND TOOIi MAKER y S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



103 



76 Deg. 


77 Deg. 


MIN 


SINK. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 




2 

4 


.97030 
.97044 
.97058 


4.0108 
4.0207 
4.0307 


.24192 
.24136 
.24079 


4.134 
4.143 
4.153 



2 

4 


.97437 
.97450 
.97463 


4.3315 
4.3430 
4.3546 


.22495 
.22438 
.22382 


4.445 
4 457 
4.468 


6 

8 
10 


.97072 
.97086 
.97100 


4.0408 
4.0509 
4.0611 


.24023 
.23966 
.23910 


4.163 
4.172 

4.182 


6 

8 
10 


.97476 
.97489 
.97502 


4.3662 
4.3779 
4.3897 


.22325 
.22268 
.22212 


4.479 
4.491 

4.502 


12 
14 
16 


.97113 

.97127 
.97141 


4.0713 
4.08.15 
4.0918 


.23853 
.23797 
.23740 


4.192 
4.202 
4.212 


12 

14 
16 


.97515 
.97528 
.97541 


4.4015 
4.4134 
4.4253 


.22155 
.22098 
.22041 


4.514 
4.525 
4.537 


18 
20 
22 


.97155 
.97169 
.97182 


4.1022 
4.1126 
4.1230 


.23684 
.23627 
.23571 


4.222 
4.232 
4.242 


18 
20 
22 


.97553 
.97566 
.97579 


4.4373 
4.4494 
4.4615 


.21985 
.21928 
.21871 


4.549 
4.560 
4.572 


24 
26 
28 


.97196 
.97210 
.97223 


4.1335 
4.1440 
4.1546 


.23514 
.23458 
.23401 


4.253 
4.263 
4.273 


24 
26 
28 


.97592 
.97604 
.97617 


4.4737 
4.4860 
4.4983 


.21814 
.21758 
.21701 


4.584 
4.596 
4.608 


30 
32 
34 


.97237 
.97251 
.97264 


4.1653 
4.1760 
4.1867 


.23345 

.23288 
.23231 


4.284 
4.294 
4.304 


30 
32 
34 


.97630 
.97642 
.97655 


4.5107 
4.5232 
4.5357 


.21644 
.21587 
.21530 


4.620 
4.632 
4.645 


36 
38 
40 


.97278 
.97291 
.97304 


4.1976 

4.2084 
4.2193 


.23175 
.23118 
.23062 


4.315 
4.326 
4.336 


36 

38 
40 


.97667 
.97680 
.97692 


4.5483 
4.5609 
4.5736 


.21474 
.21417 
.21360 


4.657 
4.669 
4.682 


42 
44 
46 


.97318 
.97331 
.97345 


4.2303 
4.2413 
4.2524 


.23005 
.22948 
.22892 


4.347 
4.358 
4.368 


42 
44 
46 


.97705 
.97717 
.97729 


4.5864 
4.5993 
4.6122 


.21303 
.21246 
.21189 


4.694 
4.707 
4.719 


48 
50 
52 


.97358 
.97371 
.97384 


4.2635 
4.2747 
4.2859 


.22835 
.22778 
.22722 


4.379 
4.390 
4.40 L 


48 
50 
52 


.97742 
.97754 
.97766 


4.6252 
4.6382 
4.6514 


.21132 
.21076 
.21019 


4.732 
4.745 
4.758 


54 
56 
6b 


.97398 
.97411 
.97424 


4.2972 
4.3086 
4.3200 


.22665 
.22608 
.22552 


4.412 
4.423 
4.434 


54 
56 
58 


.97778 
.97791 
.97803 


4.6646 
4.6779 
4.6912 


.20962 
.20905 
.20848 


4.771 

4.783 
4.797 



104 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 







78 Deg. 








79 Deg. 




WIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.97815 
.97827 
.97839 


4.7046 
4.7181 
4.7317 


.20791 
.20734 
.20677 


4.810 
4.823 
4.836 



2 

4 


.98163 

.98174 
.98185 


5.1445 
5.1606 
5.1767 


.19081 
.19024 
.19067 


5.241 
5.257 
5.272 


6 

S3 

1J 


.97851 
97863 

.97875 


4.7453 
4.7591 

4.7729 


.20620 
.20563 
.20506 


4.850 

4.863 
4.876 


6 

8 

10 


.96196 
.98207 
.98218 


5.1929 

5.2092 
5.2257 


.18909 
.18852 
.18795 


5.288 
5.304 
5.320 


12 
14 
16 


.97887 
.97899 
.97910 


4.7867 
4.8007 
4.8147 


.20449 
.20393 
.20336 


4.890 
4.904 
4.917 


12 
14 
16 


.98229 
.98240 
.98250 


5.2422 
5.2588 
5.2755 


.18738 
.18681 
.18624 


5.337 
5.353 
5.369 


38 
20 
L2 


.97922 
.97934 
.97916 


4.8288 
4.8430 
4.8573 


.20279 
.20222 
.20165 


4.931 
4.945 
4.959 


18 
20 
22 


.98261 

.98272 
.98283 


5.2923 
5.3093 
5.3263 


.18567 
.18509 
.18452 


5.386 
5.403 
5.419 


24 

26 
!28 


.97958 
.97969 
.97981 


4.8716 
4.8860 
4.9006 


.20108 
.20051 
.19994 


4.973 

4.987 
5.001 


24 
26 

28 


.98294 
.98304 
.98315 


5.3434 
5.3607 
5.3780 


.18395 
.18338 
.18281 


5.436 
5.453 
5.470 


30 
32 
34 


.97992 
.98004 
.98016 


4.9152 

4.9298 
4.9446 


.19937 
.19880 
.19823 


5.016 
5.030 
5.045 


30 
32 
34 


.98325 
.98336 
.98347 


5.3955 
5.4131 

5.4308 


.18223 
.18166 
.18109 


5.487 
5.505 
5.522 


36 

38 

40 


.98027 
.98039 
.98050 


4.9594 
4.9744 
4.9894 


.19766 
.19709 
.19652 


5.059 
5.074 
5.089 


36 
38 
40 


.98357 
.98368 
.98378 


5.4486 
5.4665 
5.4845 


.18052 
.17994 
17937 


5.540 
5.557 
5.575 


42 
44 
46 


.98061 

.98073 
.98084 


5.0045 
5.0197 
5.0350 


.19594 
.19537 
.19480 


5.103 
5.118 
5.133 


42 
44 
46 


.98389 
.98399 
.98409 


5.5026 
5.5209 
5.5393 


.17880 
.17823 
.17766 


5.593 
5.611 
5.629 


49 
6. J 

52 


.98096 
.98107 
.98118 


5.0504 
5.0658 
5.0814 


.19423 
.19366 
.19309 


5.148 
5.164 
5.179 


48 
50 
52 


.98420 
.98430 
.98440 


5.5578 
5.5764 
5.5951 


.17708 
.17651 
.17594 


5.647 
5.665 
5,684 


51 

56 
58 


.98129 
.98140 
.98152 


5.0970 
5.1128 
5.1286 


.19252 
.19195 
.19138 


5.194 
5.210 
5.225 


54 
56 
58 


.98450 
.98461 
.98471 


5.6139 
5.6329 
5.6520 


.17536 
.17479 
.17422 


5.702 
5.721 
5,740 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



105 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



80 Deg. 


81 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC- 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.98481 
.98491 
.98501 


5.6713 
5.6906 
5.7101 


.17365 
.17307 
.17250 


5.759 
5.778 
5.797 



2 

4 


.98769 

.98778 
.98787 


.63137 
.63376 
.63616 


.15643 

.15586 
.15528 


6.392 
6.416 
6.440 


6 

8 

10 


.9*511 
.98521 
.98531 


5.7297 
5.7495 
5.7694 


.17193 
.17135 
.17078 


5.816 
5.836 
5.855 


6 

8 
10 


.98796 

.98805 
.98814 


.63859 
.64103 
.64348 


.15471 
.15+13 
.15356 


6.464 

6.488 
6.512 


12 

14 
16 


■9854 L 
.985^1 
.98560 


5.7894 
5.8095 
5.8298 


.17021 
.16963 
.16906 


5.875 
5.895 
5.915 


12 
14 
16 


.98823 
.9*832 

.98840 


.64596 
.64846 
.65097 


.15298 
.15241 
.15183 


6.536 
6.561 
6.586 


18 
20 
22 


.98570 
.98580 
.98590 


5.8502 
5.8708 
5.8915 


.16849 
.16791 
.16734 


5.935 
5.955 
5.976 


18 
20 
22 


.98849 
.98858 
.98867 


.65350 
.65605 
.65863 


.15126 
.15068 
.15011 


6.611 
6.636 
6.662 


24 
26 

28 


.98599 
.98609 
.98619 


5.91-24 
5.9333 
5.9545 


.16677 
.16619 
.16562 


5.996 
6.017 
6.038 


24 
26 
28 


.98875 
.98884 
.98893 


.66122 
.66383 
.66646 


.14953 
.14896 
.14838 


6.687 
6.713 
6.739 


30 
32 
34 


.98628 
.98638 
.98647 


5.9758 
5.9972 
6X188 


.16504 
.16447 
.16390 


6.059 
6.080 
6.101 


30 
32 
34 


.98901 
.98910 
.98918 


.66912 
.67179 
.67449 


.14781 
.14723 
.14666 


6.765 
6.792 
6.818- 


36 

38 
40 


.98657 
.98666 
.98676 


6.0405 
6.0624 
6.0844 


.16332 
.16275 
.16218 


6.123 
6.144 
6.166 


36 
38 
40 


.98927 
.98935 
.98944 


.67720 
.67994 
.68269 


.14608 
.14551 
.14493 


6.845 
6.872 - 
6.900 


42 
44 
46 


.98685 
.98695 
.98704 


6.1066 
6.1290 
6.1515 


.16160 
.16103 
.16045 


6.188 
6.210 
6 232 


42 
44 
46 


.98952 
.98961 
.98969 


.68547 
.68828 
.69110 


.14435 
.14378 
.14320 


6.927 
6.955 
6.983 


48 
50 
52 


.98713 
.98723 

.98732 


6.1742 
6.1970 
6.2200 


.15988 
.15930 
.15873 


6.255 
6.277 
6.300 


48 
50 
52 


.98977 
.98986 
.98994 


.69395 
.69682 
.69972 


.14263 
.14205 
.14148 


7.011 
7.040 
7.068 


54 
56 
58 


.98741 
.98750 
.98759 


6.2432 
6.2665 
6.2901 


.15816 
.15758 
.15700 


6.323 
6.346 
6.369 


54 
56 

58 


.99002 
.99010 
.99018 


.70264 
.70558 
.70855 


.14090 
.14032 
.13975 


7.097 
7.126 
7.156 



106 



THE MACHINIST AND TOOL MAKERS' INSTRUCTOR. 



NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 



82 DEG. 


83 Deg. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.99027 
.99035 
.99043 


7.1154 
7.1455 

7.1759 


.13917 
.13859 
.13802 


7.185 
7.215 
7.245 



2 

4 


.99255 
.99262 
.99269 


8.1443 

8.1837 
8.2234 


.12187 
.12129 

.12071 


8.205 
8.245 

8.284 


6 

8 

10 


.99051 
.99059 
.99067 


7.2066 
7.2375 
7.2687 


.13744 
.13687 
.13629 


7.276 
7.306 
7.337 


6 

8 
10 


.99276 

.99283 
.99289 


8.2635 
8.3041 
8.3450 


.12014 
.11956 
.11898 


8.324 
8.364 
8.405 


12 
14 
16 


.99075 
.99082 
,99090 


7.3002 
7.3319 
7.3639 


.13571 
.13514 
.13456 


7.368 
7.400 
7.431 


12 
14 
16 


.99296 
.99303 
.99310 


8.3862 
8.4279 
8.4701 


.11840 
.11782 
.11725 


8.446 
8.487 
8.529 


18 
20 
22 


.99098 
.99106 
.99114 


7.3962 
7.4287 
7.4615 


.13398 
.13341 
.13283 


7.463 
7.496 

7.528 


18 
20 
22 


.99317 
.99324 
.99331 


8.5126 
8.5555 
8.5989 


.11667 
.11609 
.11551 


8.571 
8.614 
8.657 


24 
26 
28 


.99121 
.99129 
.99137 


7.4946 
7.5281 
7.5618 


.13225 
.13168 
.13110 


7.561 
7.594 
7.628 


24 
26 
28 


.99337 
.99344 
.99350 


8.6427 
8.6870 
8.7317 


.11494 
.11436 

.11378 


8.700 
8.744 
8.789 


30 
32 
34 


.99144 
.99152 
.99159 


7.5957 
7.6300 
7.6647 


.13052 
.12995 
.12937 


7.661 
7.695 
7.730 


30 
32 
34 


.99357 
.99363 
.99370 


8.7769 

8.8225 
8.8686 


.11320 
.11262 
.11205 


8.834 
8.879 
8.925 


36 
38 
40 


.99167 
.99174 
.99182 


7.6996 
7.7348 
7.7703 


.12879 
.12822 
.12764 


7.764 
7.799 
7.834 


36 

38 
40 


.99377 
.99383 
.99389 


8.9152 
8.9623 
9.0098 


.11147 
.11089 
.11031 


8.971 
9.018 
9.065 


42 
44 
46 


.99189 
.99197 
.99204 


7.8062 
7.8424 
7.8789 


.12706 
.12648 
.12591 


7.870 
7.906 
7.942 


42 
44 
46 


.99396 
.99402 
.99409 


9.0579 
9.1065 
9.1555 


.10973 
.10915 
.10857 


9.113 
9.161 
9.210 


48 
50 
52 


.99211 

.99219 
.99226 


7.9158 
7.9530 
7.9906 


.12533 
.12475 
.12418 


7.979 
8.016 
8.053 


48 
50 
52 


.99415 
.99421 
.99427 


9.2052 
9.2553 
9.3060 


.10800 
.10742 
.10684 


9.259 
9.309 
9.360 


54 
56 

58 


.99233 
.99240 
.99247 


8.0285 
8.0667 
8.1054 


.12360 
.12302 
.12245 


8.090 
8.128 
8.167 


54 
56 
58 


.99434 
.99440 
.99416 


9.3572 
9.4090 
9.4614 


.10626 
.10568 
.10511 


9.410 
9.462 
9.5U 





THE MACHINIST AND TOOL MAKER'S INSTRUCTOR, 


107 




NATURAL, SINES, TANGENTS 
AND SECANTS. 


, COSINES 




84 Deg. 


85 Deg. 


KIN 


SINK. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.99452 
.99458 
.99464 


9.5144 

9.5679 
9.6220 


.10453 
.10395 
.10337 


9.567 
9.620 
9.674 



2 

4 


.99619 
.99624 
.99629 


11.4301 

11.5072 
11.5853 


.08716 
.08657 
.08599 


11.474 
11550 
11.628 


6 

8 
10 


.99470 
.99476 
.99482 


9.6768 
9.7322 
9.7882 


o!0279 
.10221 
.10163 


9.728 
9.783 
9.839 


6 

8 
10 


.99634 
.99639 
.99644 


11.6645 
11.7448 
11.8262 


.08542 
.08484 
.08426 


11.707 
11.787 
11.868 


12 
14 
16 


.99458 
.99494 
.99499 


9.8448 
9.9021 
9.9601 


.10105 
.10047 
.09990 


9.895 
9.952 
10.010 


12 
14 
16 


.99649 
.99654 
.99659 


11.9087 
11.9923 
12.0772 


.08368 
.08310 
.08252 


11.950 
12.034 
12.118 


18 
20 
22 


.99505 
.9951 L 
.99517 


10.0187 
10.0780 
10.1381 


.09932 
.09874 
.09816 


10.068 
10.127 
10.187 


18 
20 
22 


.99664 
.99668 
.99673 


12.1632 
12.2505 
12.3390 


.08194 
.08136 
.08078 


12.204 
12.291 
12.379 


24 
26 
28 


.99523 
.99528 
.99534 


10.1988 
10.2602 
10.3224 


.09758 
.09700 
.09642 


10.248 
10.309 
10.371 


24 
26 
28 


.99678 
.99682 
.99687 


12.4288 
12.5199 
12.6124 


.08020 
.07962 
.07904 


12.469 
12.560 
12.652 


30 
32 
34 


.99539 
.99545 
.99551 


10.3854 
10.4491 
10.5136 


.09584 
.09526 
o09468 


10.433 
10.497 
10.561 


30 
32 
34 


.99692 
.99696 
.99701 


12.7062 
12.8014 
12.8961 


.07846 
.07788 
.07730 


12.745 
12.840 
12.937 


36 
38 

40 


.99556 
.99562 
.99567 


10.5789 
10.6450 
10.7119 


.09411 
.09353 
.09295 


10.626 
10.692 
10.758 


36 
38 
40 


.99705 
.99709 
.99714 


12.9962 
13.0958 
13.1969 


.07672 
.07614 
.07556 


13.034 
13.134 
13.235 


42 
44 
46 


.99572 
.99578 
.99583 


10.7797 
10.8483 
10.9178 


.09237 
.09179 
.09121 


10.826 
10.894 
10.963 


42 
44 
46 


.99718 
.99723 

.99727 


13.2996 
13.4039 
13.5098 


.07498 
.07440 
.07382 


13.337 
13.441 
13.547 


48 
50 
52 


.99588 
.99594 
.99599 


10.9882 
11.0594 
11.1316 


.09063 
.09005 
.08947 


11.033 
11.105 
11.176 


48 
50 
52 


.99731 
.99736 
.99740 


13.6174 

13.7267 
13.8378 


.07324 
.07266 
.07208 


13.654 
13.763 
13.874 


54 

56 

58 


.99604 
.99609 
.99614 


11.2048 
11.2789 
11.3540 


.08889 
.08831 
.08773 


11.249 
11.323 
11.398 


54 
56 
58 


.99744 
.99748 
.99752 


13.9507 
14.0655 
14.1821 


.07150 

.07091 
.07034 


13.986 
14.100 
14.217 



108 


THE MACHINIST AND TOOIi MAKER'S 


INSTRUCTOR. 




NATURAL SINES, TANGENTS, COSINES 
AND SECANTS. 


86 DEG. 


87 DEG. 


.MIN 


SINE. 


TANG. 


COSINE. 


SEC. 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.99756 
.99760 
.99764 


14.3007 
14.4212 
14.5438 


.06976 
.06917 
.06859 


14.335 
14.456 
14.578 



2 

4 


.99863 
.99866 
.99869 


19.0811 
19.2959 
19.5156 


.05234 
.05175 
.05117 


19.108 
19.322 
19.541 


6 

8 

10 


.99768 
.99772 
.99776 


14.6685 
14.7954 
14.9244 


.06801 
.06743 
.06685 


14.702 
14.829 
14.958 


6 

8 

10 


.99872 
.99875 
.99877 


19.7403 
19.9702 
20.2056 


.05059 
.05001 
.04943 


19.766 
19.995 
20.230 


12 
14 
16 


.99780 
.99784 
.99788 


15.0557 
15.1893 
15.3254 


.06627 
.06569 
.06511 


15.089 
15.222 
15.35y 


12 
14 
16 


.99880 

.99883 
.99886 


20.4465 
20.6932 
20.9460 


.04885 
.04827 
.04768 


20.471 
20.717 
20.970 


18 
20 
22 


.99791 

.99795 
.99799 


15.4638 
15.6048 
15.7483 


.06453 
.06395 
.06337 


15.496 
15.637 
15.780 


18 
20 
22 


.99889 
.99892 
.99894 


21.2049 
21.4704 
21.7426 


.04711 
.04652 
.04594 


21.228 
21.494 
21.765 


24 
26 
28 


.99803 
.99806 
.99810 


15.8945 
16.0435 
16.1952 


.06279 
.06221 
.06163 


15.926 
16.075 
16.226 


24 
26 
28 


.99897 
.99900 
.99902 


22.0217 
22.3081 
22.6020 


.04536 
.04478 
.04420 


22.044 
22.330 
22.624 


30 
32 
34 


.99813 
.99817 
.99820 


16.3499 
16.5075 
16.6681 


.06104 
.06047 
.05989 


16.380 
16.538 
16.698 


30 
32 
34 


.99905 
.99907 
.99909 


22.9038 
23.2137 
23.5321 


.04362 
.04304 
.04245 


22.925 
23.235 
23.553 


36 

38 
40 


.99824 
.99827 
.99831 


16.8319 
16.9990 
17.1693 


.05930 
.05872 
.05814 


16.861 
17.028 
17.198 


36 

38 
40 


.99912 
.99915 
.99917 


23.8593 
24.1957 
24.5418 


.04187 
.04129 
.04071 


23.880 
24.216 
24.562 


42 
44 
46 


.99834 
.99837 
.99841 


17.3432 

17.5205 
17.7035 


.05756 
.05698 
.05640 


17.372 
17.549 
17.730 


42 
44 
46 


.99919 
.99922 
.99924 


24.8978 
25.2644 
25.6418 


.04013 
.03955 
.03897 


24.918 

25.284 
25.661 


48 
50 
52 


.99844 
.99847 
.99850 


17.8863 
18.0750 
18.2677 


.05582 
.05524 
.05466 


17.914 

18.103 
18.295 


48 
50 
52 


.99926 
.99928 
.99931 


26.0307 
26.4316 
26.8450 


.03839 
.03780 
.03722 


26.050 
26.450 
26.864 


54 
56 

58 


.99853 

.99857 
.99859 


18.4645 
18.6656 
18.8711 


.05408 
.05349 
.05292 


18.491 
18.692 
18.897 


54 
56 

58 


.99934 
.99^35 
.99937 


27.2715 
27.7117 
28.1664 


.03664 
.03606 
.03548 


27.290 
27.730 
28.184 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



109 



NATURAL, SINES, TANGENTS, COSINES 
AND SECANTS. 



88 Deg. 


89 Deg. 


MIS 


SINE. 


TANG. 


COSINE. 


SEC* 


MIN 


SINE. 


TANG. 


COSINE. 


SEC. 



2 

4 


.99939 
.99941 
.99943 


28.636 
29.122 
29.624 


.03490 
.03432 
.03374 


28.654 
29.139 
29.641 



2 

4 


.99985 
•99986 
.99987 


57.290 
59.266 
61.383 


.01745 
.01687 
.01629 


57.298 
59 274 
61.390 


6 

8 
10 


.99945 
.99947 
.99949 


30.145 
30.683 
31.242 


.03315 
.03257 
.03199 


30.161 

30.698 
31.257 


6 

8 
10 


.99988 
.99988 
.99989 


63.657 
66.106 
68.750 


.01571 
.01513 
.01454 


63.664 
66.112 
68.757 


12 
14 
16 


.99951 
.99952 
.99954 


31.820 
32.421 
33.045 


.03141 
.03083 
.03025 


31.836 
32.437 
33.060 


12 
14 
16 


.99990 
.99991 
.99992 


71.615 
74.729 
78.126 


.01396 
.01338 
.01279 


71.622 
74.736 
78.133 


18 
20 
22 


.99956 
.99958 
.99959 


33.694 
34.368 
35.069 


.02966 
.02908 
.02850 


33.708 
34.382 
35.084 


18 
20 
22 


.99992 
.99993 
.99994 


81.847 
85.940 
90.463 


.01222 
.01163 
.01105 


81.833 
85.984 
92.469 


24 
26 
28 


.99961 
.99962 
.99964 


35.801 
36.563 
37.358 


.02792 
.02734 
.02676 


35.814 
36.576 
37.371 


24 
26 
28 


.99995 
.99995 
.99996 


95.489 
101.107 
107.43 


.01047 
.00989 
.00931 


95.495 
101.112 
107.411 


30 
32 
34 


.99966 
.99967 
.99969 


38.188 
39.057 
39.965 


.02617 
.02560 
.02501 


38.201 
39.069 
39.978 


30 
32 
34 


.99996 
.99997 
.99997 


114.59 

122.78 
132.22 


.00872 
.00814 
.00756 


114.548 
122.850 
132.275 


36 
38 
40 


.99970 
.99971 
.99973 


40.917 
41.916 

42.964 


.02443 
.02385 
.02327 


40.931 
40.928 
42.976 


36 

38 
40 


.99997 
.99998 
.99998 


143.24 

156.26 
171.89 


.00698 
.00640 
.00582 


143.266 
156.250 
171.821 


42 
44 
46 


.99974 
.99975 
.99977 


44.066 
45.226 
46.449 


.02269 
.02210 
.02152 


44.077 
45.237 
46.460 


42 
44 
46 


.99998 
.99999 
.99999 


190.98 
214.86 
245.55 


.00524 
.00465 
.00407 


190.840 

215.05 

245.70 


48 
50 
52 


.99978 
.99979 
.99980 


47.739 
49.104 
50.548 


.02094 
.02036 
.01978 


47.750 
49.114 
50.559 


48 
50 
52 


.99999 
.99999 
.99999 


286.48 
343.78 

429.72 


.00349 
.00291 
.00233 


286.53 
343.65 
429.20 


54 
56 

58 


.99982 
.99983 
.99984 


52.081 
53.709 
55.441 


.01919 
.01861 
.01803 


52.090 
53.718 
55.450 


54 
56 
58 


.99999 

1. 

1. 


572.96 
859.44 
1718.87 


.00174 
.00116 
.00058 


574.71 

862.0 

1718.8 



90 Deg. 

Sine — 1. 
Tangent — Infinite. 
Cosine — 0. 
Secant = Infinite. 
The preceding Table of Sines, Tangents, Cosines and Secants is given 
for every two minutes of the Quadrant to a Radius of 1. 



110 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CHAPTER II. 



GEARING. 

In the transfer of motion from one axis to another gear- 
ing is the most important system that can be introduced, 
whether the shafts are to be parallel or at an angle to each 
other. 

For this purpose when the shafts are parallel, we use spur 
gears, so called from their teeth being parallel with their axis. 
*■■ When the gears are not of the same size, the smaller one 
is usually called a pinion, although the name u spur gear " is 
also correct. 

Bevel gears are used for transmitting motion between a 
pair of shafts that are at right angles to each other, and one 
wheel is always larger than the other. 

Mitre gears are so called because the pitch line of their 
teeth are always at an angle of 45° from their axis. 
When we say a pair of mitre gears, we know that they are 
both alike, that the same cutter will cut both gears, and also 
that their shafts are to be at right angles to each other. 

Thus it can be seen that a mitre is a bevel wheel (although 
we do not usually call it such), but a bevel wheel is never a 
mitre, except as stated that the pitch line is at an angle of 45° 
to its axis (or bore. ) 

Angle gears are also bevel gears, but they are so called 
from the fact that they are usually employed for transmitting 
motion between shafting that is less or greater than a right 
angle or 90°. 

Internal gears also have their teeth, like spur gears, par- 
allel to its axis, but the teeth of the internal gear are inside of 
the rim, while those of the spur gear are outside. 

Spiral gears are so called from having their teeth formed 
like a screw of a very coarse pitch, and for this reason are 



THE MACHINI8T AND TOOL MAKER'S INSTRUCTOR. Ill 

sometimes called screw gears. A spiral gear, like a spiral 
milling cutter, runs smoother than with straight teeth, and 
sometimes can be used to greater advantage than a worm 
and worm wheel. 

When a pair of spiral gears are required to connect a pair 
of shafts that are parallel to each other, one should be right 
hand and the other left hand; they may be one degree or 
twenty degrees, to suit, and whatever angle is given to one 
must also be given to the other, regardless of their size, but 
the pitch of the spiral will be in proportion to the number of 
teeth; thus in a spiral gear of twenty teeth 10 degrees angle 
and a certain pitch, another gear to mesh with it of 40 or 60 
teeth, should also be cut at 10 degrees, but the pitch would be 
twice or three times as great, or in proportion to the number 
of their teeth. 

If a pair of spiral gears are to connect at right angles to 
each other, then both wheels will be right hand or left hand, 
according to the direction they are to drive. 

In spur gears when we say a 4-inch, an 8-inch or a 20-inch 
gear, it is always understood to be the pitch diameter, and if 
these gears were of 8 pitch then their diameters would be 
4|, 8f and 20f respectively; but if they were to be 10 pitch, 
then instead of adding eighths, you would add tenths, or if 
they were to be 2 pitch you would add J^ // > or if 1 pitch you 
would then add inches. Thus a gear of 1 pitch, 20 teeth, 
would be 20 inches diameter on the pitch line and 22 inches 
outside diameter, or if you want a pair of gears, say 24 teeth 
and 30 teeth of six pitch, then the pair would be 2 / and 3 g° re- 
spectively, or 4 inches and 5 inches diameter respectively on 
the pitch line or 4f and 5f inches outside diameter. In other 
words, the gear will measure on the pitch line as many 8ths, 
lOths or 32ds as there are teeth; thus, in a wheel of 20 teeth, 8 
pitch 20/8ths ; 20 teeth 10 pitch, 20/10ths; 20 teeth 32 pitch, 
20/32ds, etc. of inches, and two more eighths should be added 
if of 8 pitch, or 2/32 more if of 32 pitch, etc. This is called 
diametral pitch and means so many teeth per inch in diam- 
eter, measured on the pitch line. 



112 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

Nearly all gears are now made on this system, for the 
reason that in building machinery the distances between two 
or more shafts that are to be driven by gears will always be 
so many 8ths apart if 8 pitch, or so many lOths, if 10 pitch, 
etc., while in the old system of circular pitch it is common to 
have the distances so many inches and a sixty-fourth full or a 
thirty-second scant, etc. 

The clearance at the bottoms of teeth should always be 
one-tenth (1/10) of the thickness of the tooth on the pitch 
line. 

To find the thickness of a tooth on the pitch line, divide 
the circumference of the wheel on the pitch line by the num- 
ber of teeth and spaces, or by twice the number of teeth, and 
one-tenth of this will be the depth of clearance at the bottom 
of the teeth. 

Thus, to find the thickness of an 8 pitch tooth, as there 
are eight teeth to every inch in the diameter measured on the 
pitch line, then a wheel of one inch diameter = 3.1416" in 
circumference, and divided by 16 = .196" for the thickness, 
and one-tenth of this for clearance = .020 nearly, for any 8 
pitch gear. 

If this was a 6 pitch gear, we would divide by 12 ; 
thus, 3.1416 -T- 12 = .262" nearly, for the thickness of any 6 
pitch tooth, and this divided by 10 = .026" for the depth of 
the clearance of any 6 pitch gear. 

If it was a 30 pitch gear, then 3.1416 -~ 60 = .052" for the 
thickness of the tooth on the pitch line, and one-tenth of this 
= .005" for the depth of the clearance, etc. 

In any gear of 8 pitch the depth of tooth is always one- 
eighth above the pitch line and one-eighth below the pitch 
line, and also the clearance ; then 2/8 = .250 and added to 
.020 = .270" for the total depth of any 8 pitch gear. 

In a 6 pitch gear the depth of tooth is one-sixth of an inch 
above the pitch line and one-sixth of an inch below the pitch 
line, plus the clearance ; thus we have explained that the 
clearance for a 6 pitch gear was .026", then added to 2/6 of an 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 113 

inch = .359" for the depth of the teeth for any 6 pitch gear, 
etc. 

It must be remembered that in all cut gearing the width 
of space on the pitch line should always be the same as the 
thickness of the teeth, and consequently, the circular pitch of 
any gear is equal to twice the thickness of the tooth on the 
pitch line. Or, dividing the circumference of the gear on the 
pitch line by the number of teeth will be the same thing. 

Thus, a wheel one inch diameter on the pitch line 
would be 3.1416 // in circumference, and if it was to be of 8 
pitch, then there would be 8 teeth on the wheel; then 
3.141 6 -r- 8 = .393" nearly, for the circular pitch, or, if it was 
10 pitch, then 3.1416 -r- 10 = .314" for the circular pitch, etc. 

Remember, that in calculating the dimensions of gear 
teeth, as in the two examples just shown, as for instance 
dividing 3.1416 by 10, it is not necessary to go higher than 
thousandths, or three decimals, and we always take for the 
last figure the one nearest to a whole number. 

Or we can find the diametral pitch in the following man- 
ner when the circular pitch is known : 

The circular pitch of a gear is %"; what is the diametral 
pitch? y s " = .875" and 3.1416 -r- .875 = 3.59, the answer. 

In other words 3.1416 divided by the circular pitch reduced 
to a decimal number, will be the diametral pitch. 

If we have a pair of gears that are 12" and 8" respect- 
ively, pitch diameters, we know that the distance between 
centers will be equal to one-half of the diameters of the two 
gears; or, 12 + 8 = 20, and 20 -f- 2 = 10". Or we may have the 
two gears like the following example : Find the distance be- 
tween centers of two gears in mesh of 28 and 24 teeth respect- 
ively and 32 pitch. 28 + 24 = 52, and 52 -r- 32 = 1% or 1.625; 
this divided by 2 = .8125, the answer. 

Suppose we have a pair of gears of 26 and 32 teeth respect- 
ively and 8 pitch, what is the distance between centers of 
these wheels when in mesh ? 26 + 32 = 58, and 58 -=- 8 pitch 
= 7.25", and this again divided by 2 = 3.625", the answer. 

Now let us examine a pair of gears in circular pitch. We 



114 



THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 



GEARING.— TOOTH DIMENSIONS. 







Thickness 








Diametral 


Circular 


of Tooth on 


Depth to 


Working 


Clearance. 


Pitch. 


Pitch. 


the Pitch 


Cut. 


Depth. 






Iyine. 








1 


3.1416 


1.5708 


2.157 


2.000 


.157 


VA 


2.5132 


1.2566 


1.725 


1.600 


.125 


1J* 


2.0944 


1.0472 


1.437 


1.333 


.104 


1% 


1.7952 


.8976 


1.232 


1.143 


.089 


2 


1.5708 


.7854 


1.078 


1.0 


.078 


2M 


1.3962 


.6981 


.958 


.888 


.070 


2^ 


1.2566 


.6283 


.863 


.800 


.063 


2% 


1.1423 


.5711 


.784 


.727 


.057 


3 


1.0470 


.5235 


.719 


M6 


.053 


3^ 


.9666 


.4833 


.664 


.616 


.048 


sy 2 


.8976 


.4488 


.616 


.571 


.045 


4 


.7854 


.3927 


.539 


.500 


.039 


5 


.6283 


.3141 


.431 


.400 


.031 


6 


.5236 


.2618 


.359 


.333 


.026 


7 


.4488 


.2244 


.308 


.286 


.022 


8 


.3927 


.1964 


.270 


.250 


.020 


9 


.3490 


.1745 


.240 


.222 


.018 


10 


.3141 


.1570 


.216 


.200 


.016 


11 


.2856 


.1428 


.196 


.182 


.014 


12 


.2618 


.1309 


.180 


.167 


.013 


14 


.2244 


.1122 


.154 


.143 


.011 


16 


.1963 


.0982 


.135 


.125 


.010 


18 


.1745 


.0872 


.120 


.111 


.009 


20 


.1570 


.0785 


.108 


.100 


.008 


22 


.1428 


.0714 


.098 


.001 


.007 


24 


.1309 


.0654 


.090 


.083 


.007 


26 


.1208 


.0604 


.083 


.077 


.006 


28 


.1122 


.0561 


.077 


.071 


.006 


30 


.1047 


.0523 


.072 


.066 


.006 


32 


.0982 


.0491 


.067 


.062 


.005 


34 


.0924 


.0462 


.063 


.059 


.004 


36 


.0872 


.0436 


.060 


.055 


.004 


40 


.0785 


.0392 


.054 


.050 


.004 


42 


.0748 


.0374 


.051 


.048 


.003 


46 


.0683 


.0341 


.047 


.043 


.004 


50 


.0628 


.0314 


.043 


.040 


.003 


54 


.0581 


.0290 


.040 


.037 


.003 


60 


.0523 


.0261 


.036 


.033 


.003 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 115 

have a pair of gears of 24 and 30 teeth and % pitch ; what is 
the distance between the centers of these two wheels when in 
mesh? 

24 X %" (or .375") = 9.000", the circumference on the 
pitch line; 9-^-3.1416 = 2.864+, the diameter on the pitch 
line. 30 X Y&" (or .375") = 11.250", or the circumference on 
the pitch line, and 11.250 -r- 3.1416 =3.581, nearly, for the 
diameter on the pitch line ; and 2.864" + 3.581" = 6.445", 
then divided by 2 = 3.2225 inches, or the answer. 

This can also be found in a more simple manner as fol- 
lows : Multiply the constant number .3183 by the circular 
pitch, and the result by one-half the number of teeth in both 
wheels; thus, in the same example, .3183 X .375 = .11936+, 
and one-half the number of teeth in both wheels = 27; then 
.11936 X 27 = 3.222", the answer, or the distance between the 
centers of the two wheels. Any example in circular pitch can 
be done in the same manner. 

The decimal .3183 is found by dividing one by 3.1416. 

The depth of teeth in circular pitch is found in the fol- 
lowing manner: Thus, for a gear of J^ 7/ circular pitch 
3.1416 divided by .5 {%) = 6.283, for the diametral pitch, and 1 
divided by 6.283 = .159", for one -half of the running depth 
of the tooth, and twice this, or, .318" = the running depth. 
Now if the circular pitch is 3^3", then the tooth is exactly J4" 
thick on the pitch line, and the clearance we said was one- 
tenth (1/10) of the thickness of the tooth on the pitch line; 
then 1/10 of J£, (or .250") = .025" and added to .318" = .343" 
for the depth to cut any gear of 3^ 7/ circular pitch. 

We can also find the depth of the teeth in a more simple 
manner, thus : Multiply the number (constant) .3183 by the 
circular pitch and this will give one-half of the running depth 
of the tooth; thus, in y 2 " circular pitch 3^" = .5"; .3183 
X .5 = .159, and doubled equals .318, and one-tenth of the 
thickness of the tooth = .343", the answer, or total depth as 
before. 

It can readily be seen that if we want to have a pair of 
gears in mesh, and the holes are bored to the proper distance 



116 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

for standard gears, and we then take a pair of gears that are 8 
or 10 thousands too large or too small in the diameter, it will 
be too tight or too loose, and if the gear has 15 teeth and 
should be ten thousandths of an inch too small in the diam- 
eter, then it would be about 30 thousandths of an inch too 
small or to be divided into the 15 teeth, or about two thou- 
sandths of an inch for each tooth, etc. 

In a great many instances this would never be known, and 
especially in a rough class of work, such as rolling mill ma- 
chinery, etc. But in a fine class of work the greatest care 
should be taken to have the blanks turned to an exact size by 
means of the micrometer caliper or the vernier, as the case 
may be, and then if the gear cutter or milling machine is 
graduated correctly (some are not) we will have no trouble in 
getting good work. 

In designing the shapes of teeth, it is well to remember 
that the teeth of a pair of wheels should always be in contact 
at least equal to the pitch. Or, in other words, if we have a 
pair of wheels in mesh with each other of 3 7/ pitch, the teeth 
should not separate from each other until the next pair of 
teeth are well in contact, otherwise the friction from the slid- 
ing motion will be very great and a uniform motion cannot be 
maintained. 

It has also been demonstrated by practice that any 
describing curve rolled on the outside of one pitch line and 
on the inside of another will work correctly together. 

An epicycloid is somewhat similar, except that it is traced 
by a point on the circumference of a circle, as shown in 
figure 7, rolling upon the pitch line of another, and, when 
rolling on the inside it is called a hypocycloid. The circle C 
is called the generating circle, because it generates (or forms) 
the curves for the shapes of the teeth. 

This generating circle is usually made equal to the radius 
(half the diameter) of the smallest wheel in a train of gearing. 
Suppose this smallest wheel to be 3 pitch, 15 teeth or 5 inches 
in diameter, then the generating circle would be V/%" diameter. 
This same circle would be used for all the wheels con- 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 117 

necting in a train, rolled on the outside of the pitch line (fig- 
ure 7) for generating the faces of the teeth, and on the inside 
(Figure 8) for the flanks of the teeth. 

Now if we roll this ^Yz' generating circle on the inside of 
a 5" pitch circle, we will have a perfectly straight line from 
the pitch line through the center of the gear, or, as we call 
them, radial lines, for the flanks of the teeth of this 5 /7 gear. 

It is not necessary that this size of generating circle 
should be used for this 5" gear, but it is the size best adapted 
for general purposes. 

If we use a generating circle that is less than one-half the 
diameter of the pitch circle, the flanks of the teeth will be 
more nearly parallel with each other, while the faces of the 
teeth will be more pointed than before, and the latter will 
naturally make the teeth crowd harder on each other, and, at 
the same time, if the flanks of the teeth are parallel, even fo r 
a short distance, the cutters will work hard, as there can be 
no clearance on the teeth of cutters when parallel. (This re- 
fers to formed cutters only.) 

In the epicycloidal form of gear, one of the most simple, 
is shown at Figure 1, and is usually called pin gearing, so called 
because one of the gears is made of two discs a little larger 
than the outside of the pins which are inserted for the teeth, 
as shown at C, this wheel is made somewhat wider than the 
large wheel, otherwise they could not run together. 

It must not be forgotten that the pitch lines G G, of both 
wheels must be in proportion to the numbers of teeth in the 
two wheels; thus, if the large wheel has 14 teeth, and the small 
one 7 teeth (or pins), the pitch diameter of one wheel must be 
just twice as great as the other. 

In designing a pair of wheels of this kind we will suppose 
the pins C to be % /f diameter ; the distance between two 
holes on the pitch line should be % /7 or % circular pitch, and, 
if there were 7 pins in the small wheel, then the circumference 
of the pitch circle G would be 7 times %" around it ; this will, 
of course, make the teeth on the large wheel %" thick on the 
pitch line also. 



118 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



To draw the teeth on the large wheel, first lay them out 
just %f f between centers on the pitch circle as shown, and 
draw the circles equal to the size of the pins for the bottom of 
the teeth, and, with the dividers set in one of these circles, H, 
and the other leg of the dividers touching the edge of the next 
circle, draw the addendum (or that part of the tooth outside 
the pitch circle) shown at I, etc. 



» 






JF 









- ^*-/5» 



av 



1 1/ 



»i; 






73 

i 

i 



mt 



l/B 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



119 



Figure 2 represents the single curve gear, and is 
of 4 pitch (diametral) 13 teeth, and consequently is 
13-4ths or 3 \^ ,f diameter on the pitch line A A, which 
should be drawn first, and as any 4 pitch gear is .393 7/ 
thick on the pitch line, lay out with the dividers several points 
equal to this distance on the line A A ; then with bevel pro- 
Pi > 




tractor, or what is better, a piece of thin sheet steel cut to an 
angle of 75° 30', and placing the corner of this template on the 
pitch line at one of these points as shown, and with one edge 
in line with the center, measure off on the other side from the 
corner of the template a distance equal to one-fourth (34) of 



120 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

the radius, or from the pitch line to the center of the gear, 
which in this case is about .406 7/ , as shown at B, and, with the 
dividers set in the center of this gear describe at B a circle, 
which is called the base circle, shown dotted in the figure. 
Having drawn the base circle, we do not have to use this tem- 
plate again on this gear for the reason that having spaced off 
on the pitch circle (not the base circle) as directed, and with 
the dividers set to .406 // as nearly as possible, we place one leg 
of the dividers on the base circle and the other on the pitch 
circle at one of these points stepped off and draw the curves 
for the faces of the teeth to the base circle only. 

Now it has been found that for 12 and 13 teeth that paral- 
lel lines should be used for that portion of the teeth inside of 
the base circle, and drawing a circle at the center of the gear, 
as shown, for convenience, equal to the width of space at the 
base circle, draw lines C C, and for the fillets at the bottom of 
the teeth, the radius should be about one-sixth (1/6) of the 
distance B B. With gears having 20 teeth the curves may ex- 
tend from the addendum to the fillets at the bottom (or two 
pitches deep.) 

In all gears from 13 to 20 teeth, there should be parallel 
lines inside of the base circle for a portion of the distance. 
With a little practice in drawing the teeth of a pair of wheels 
in mesh with each other, and in different positions, making 
the lines as fine and accurate as possible, and also practicing 
on a wheel of very coarse pitch, you will soon learn how to 
form the inside or flanks of the teeth. Remember the outside 
should be made as described, and the inside, or that portion 
inside of the base circle to suit the outside or addendum of 
the teeth. 

If you want to make a drawing at any time, say of a pair 
of gears of any pitch (in order to find the shapes of the teeth), 
draw the teeth as stated, in different positions, and make them 
five or ten times (for convenience) as large as required; this 
will be on the principle of a magnifying glass, and will show 
the errors very plainly; this refers to gears of any pitch, style, 
etc. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 121 

Remember that a gear of 60 pitch and any number of teeth 
is of the same shape as a gear of 2 pitch (or any other pitch) 
with the same number of teeth, and, of course, using the same 
Style of a tooth. 

That is, an involute tooth, of 32 pitch 15 teeth, is of the 
same shape as an involute tooth of 1 pitch 15 teeth; the only 
difference is, that one is much smaller or finer than the other. 

I have had a great many cutters made by making the 
drawings of three or four teeth of a gear on smooth stiff paper 
and then cutting out with a sharp knife ; but when regular 
standard cutters are wanted, it is cheaper and better to get 
them from parties who make a business of this class of work. 

In Figure 3, I have shown a double curve gear of Y^ n 
pitch, 40 teeth. There are no fillets shown at the bottom of 
these teeth, because they are strong enough without them ; in 
gears with a small number of teeth, however, it will be neces- 
sary to have fillets the same as in a single curve. 

This style of gear teeth was invented by Professor Willis, 
after a great deal of experimenting, and very nearly approach- 
es a perfect epicycloid. It is drawn in the following manner 

The example is 40 teeth }/%" pitch ; therefore, 40 times 
y%' around on the pitch line, or 20 //f circumference, and 
divided by 3.1416= 6.366 inches in diameter, on the pitch line 
A A, which should be drawn first ; as the gear is J£ inch 
pitch, the tooth should be exactly \^ ,r thick on this line. Now 
having made several points on the pitch circle A A, y±" from 
each other (for }/%" pitch, or 5/16" for %" pitch), with a bevel 
protractor, or what is better, a piece of thin sheet steel for a 
template, as shown at B C D of 75° placed at one of these 
points; we find in the table (that follows) for 40 teeth, J^ /7 
pitch, the number 4 ; this means 4/20 of an inch, and is for 
the face of the teeth; then with the dividers set to 4/20 to the 
right of the corner of the template, make the point F, and 
from this point lay off the face of the tooth as shown by the 
small circle. 



122 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




■<- 



\JE/ 



75 



For the flank, or inside of the same tooth, we move the 
instrument one pitch, whatever it may be; in this case it is 
y%" to the left, as shown, and in the table for the flanks of 
teeth we find the number 7 for 40 teeth y% f pitch, and we now 
measure off 7/20" to the left of the corner, and from this 
point describe the curve for the flank, as shown by the large 
circle. 

It must be remembered that the numbers in the table are 
so many 20ths of inches. If you want to draw several of these 



TABLE FOR DOUBLE CURVE GEAR TEETH. 

(wims SYSTEM.) 
CENTERS FOR THE FLANKS OF THE TEETH. 

PITCH IN INCHES AND PARTS. 



NUMBER 
TEETH. 


Y2 


% 


% 


1 


im m 


i 


2 


2M 2^ 


3 \m 


13 


64 


80 


96 


129 


160 


193 


225 


257 


289 


321 


3a6 450 


14 


35 


43 


52 


69 


87 


104 


121 


139 


156 


173 


208 


242 


15 


25 


31 


37 


49 


62 


74 


86 


99 


111 


123 


148 


17a 


16 


20 


25 


30 


40 


50 


59 


69 


79 


89 


99 


191 


138 


17 


17 


21 


25 


34 


42 


50 


59 


67 


75 


84 


101 


117 


18 


15 


19 


22 


30 


37 


45 


52 


59 


67 


74 


89 


104 


19 


13 


17 


20 


27 


35 


40 


47 


54 


60 


67 


80 


94 


20 


12 


16 


19 


25 


31 


37 


43 


49 


56 


62 


74 


86 


22 


11 


14 


16 


22 


27 


33 


39 


43 


49 


54 


65 


76 


24 


10 


12 


15 


20 


25 


30 


35 


40 


45 


49 


59 


69 


26 


9 


11 


14 


18 


23 


27 


32 


37 


41 


46 


55 


64 


28 






13 




22 


26 


30 


35 


40 


43 


52 


60 


30 


8 


10 


12 


17 


21 


25 


29 


33 


37 


41 


49 


58 


35 




9 


11 


16 


19 


23 


26 


30 


34 


38 


45 


53 


40 


7 






15 


18 


21 


25 


28 


32 


35 


42 


49 


60 


6 


8 


9 


13 


15 


19 


22 


25 


28 


31 


37 


43 


80 




7 




12 




17 


20 


23 


26 


29 


35 


41 


100 






8 


11 


14 






22 


25 


28 


34 


3& 


150 


5 








13 


16 


19 


21 


24 


27 


32 


38 


Rack, 




6 


7 


10 


12 


15 


17 


20 


22 


25 


30 


34 



CENTERS FOR THE FACES OF THE TEETH. 

PITCH IN INCHES AND PARTS. 



NUMBERS 

OF 

TEETH. 


H 


% 


% 


1 


iM 


IJfi 


1% 


2 


m 


2^ 


3 


3^ 


12 


2 


3 


4 


5 


6 


7 


9 


10 


11 


12 


15 


17 


15 


3 








7 


8 


10 


11 


12 


14 


17 


19 


20 




4 


5 


6 


8 


9 


11 


12 


14 


15 


18 


21 


30 


4 






7 


9 


10 


12 


14 


16 


18 


21 


25 


40 






6 


8 




11 


13 


15 


17 


19 


23 


2a 


60 




5 






10 


12 


14 


16 


18 


20 


25 


29 


80 








9 


11 


13 


15 


17 


19 


21 


26 


30 


100 






7 










18 


20 


22 




31 


150 


5 


6 








14 


16 


19 


21 


23 


27 


32 


Rack. 








10 


12 


15 


17 


20 


22 


25 


30 


34 



For 4" Pitch, double, 2"; For 5" Pitch, double, 2J£", etc. 



124 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

teeth, then with one leg of the dividers set to the center of 
the gear at I, draw arcs through the centers of these two cir- 
cles shown in the figure, and you will not have to use the 
template again on this gear. 

Figure 4 represents a perfect epicycloidal tooth, both on 
the pinion and the rack; they are of 4 pitch, and the pinion 
has 15 teeth, and were drawn in the following manner: For 
the pinion, turn up a block of wood with a radius equal to the 
radius of the pinion on the pitch line, also another piece 
(concave) to match the other. Now take the dividers and lay 
off the pitch circle on paper, and place one of these wood 
blocks (made as thin as possible) and with another round 
block turned up just one-half the diameter of the pitch circle 
of the pinion, and, holding a lead pencil close to the periphery, 
and rolling it on this circle, describe the curves, first on one 
side and then the other (inside and outside of the pitch circle); 
these curves will form a perfect epicycloid. 

One straight block of wood will answer for the rack, plac- 
ing it on the pitch line G D, and rolling the same block first 
on one side and then the other as before. 

Two sizes of rolls can be used on a pair of gears, but what- 
ever size is used for the outside, or face, of one gear must, in 
all cases, be used for the inside of its mate. 

This rack and pinion can also be done in the same man- 
ner as described in the double curve gears; thus, for the rack, 
place the instrument with the corner touching the pitch line 
as shown to the left of the figure, and, as there is no center 
to a rack as in the pinion, the end A will have to be placed at 
90° to the pitch line ; in the table find the number correspond- 
ing to the pitch required, and describe the curve for the face, 
and then move one pitch to the left, and describe the flank 
as already explained. 

Figure 5 also shows a pinion of 15 teeth in mesh with an 
internal gear of 30 teeth, and was drawn in the same manner 
as in Figure 4. It will be seen that the teeth of the internal 
wheel are similar in shape to that of the rack, except that in 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 125 




126 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 12? 

the internal wheel the teeth are thicker at the root than in 
the rack, this being due to the curve of the wheel. 

An internal wheel can be drawn in the same manner as 
any other gear, the difference being that the teeth as drawn 
becomes the space and the space the teeth. 

In figure 6 I have shown the manner of constructing the 
perfect involute shape of gear teeth, which is as follows : 
From the center D, describe the pitch circle A (a small por- 
tion only being seen) and on this circle step off points for the 
teeth as already explained ; we then place our template or pro- 
tractor of 75° 30 / with the corner on the pitch line A; and the 
edge in line with the center of the gear, measuring off one- 
fourth of the radius of the gear, which, in this example, is ^ 7/ 
(the radius being 3 // ). We now have a point from which 
with the dividers set (the other point in the center at D) we 
describe the arc C B, which is called the base circle. 

Remember that the involute always extends from the 
base circle, and not the pitch circle, to the top of the tooth, 
while that portion of the tooth inside the base circle is the 
same as explained in single curve gears. We then lay out a 
number of radial lines, extending from the base line to the 
center ; it is not necessary to start from any particular place 
in the figure. I have started nearly %" from the side of the 
tooth. I could have made it %"; the more lines and the 
closer they are, if properly drawn, the more correct will be 
the tooth, but it can easily be seen that I have made them 
close enough for all purposes ; but whatever distance you 
make 1 and 2, make the rest the same, or in other words, space 
them evenly. Next from these radial lines draw the lines 
marked V 2 / 3', etc., exactly at 90° from the radial lines, and, 
with the dividers set with one point at 1 on the base line, and 
the other point at F, also on the base line, draw that portion 
of the tooth between F and 2', then open the dividers, and 
placing one leg at 2 on the base circle, and with the pencil at 
2 / extend the tooth to 3 7 , then open again, and placing one 
point of the dividers at 3 on the base circle, and the pencil at 
3' extend the tooth to 4', etc. Continue in this w r ay until you 



128 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL. MAKEK'S INSTRUCTOR. 129 

finish the tooth as shown. It can easily be seen by the draw- 
ing that a single arc struck from any position will not be an 
involute. 

The true involute tooth is the best by far for the shapes 
of gear teeth for most purposes, for the reason that when the 
distance between centers are not correct it will make no dif- 
ference in their uniformity of speed, etc* 

Figure 7 illustrates a method of drawing the perfect 
epicycloid, as follows : B is the center of the gear B A, the 
pitch circle, and C the generating circle, which in this case I 
have made a little less in diameter than the radius of the gear, 
in order to show a curve in the next figure, as both these 
figures 7 and 8 show but one side of a tooth ; figure 7 the face 
of a tooth, and figure 8 the flank of the same tooth. 

In Figure 7 draw radial lines from the center B, through 
the pitch circle A B, as shown; they must be produced fa 
enough to reach the top of the tooth. Make these lines uni- 
form in their distance one from the other; it is not important 
what the distance is, but do not get them too far apart. Sup- 
pose them to be one-eighth of an inch starting from A, toward 
B; then on the generating circle C, also commencing at A, 
step off points exactly the same distance J£ 7/ , as shown at 1, 2, 
3, etc., and with the dividers set to the center of the circle at 
B, draw arcs from these points on the generating circle, as 
shown at 1, 2, 3, etc. Remember to draw the arc from the 
line F B to 1 on the generating circle ; next draw the arc from 
the line F B to 2 on the generating circle ; in the same man- 
ner draw all the arcs, and as there are 13 points on the gener- 
ating circle, then the 13th arc will be continued through to 
the 13th radial line. 

Now to find the shape of the face of the tooth A D above 
the pitch line, place one leg of the dividers on the 13 point on 
the generating circle near I and the other leg at G, then re- 
move the dividers to H, and mark the point I; next with one 
leg of the dividers in the 12 space on the generating circle 
and the other leg at J, remove as before to K, and describe 
the point shown at L. Continue in this way until you pro- 



130 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR 



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duce all the points shown on the arc A D, and a curve through 
these points will be an epicycloid. 

Figure 8 shows the same system for drawing the flanks of 
the same tooth. B is the center of the gear, A E is the pitch 
circle, and C the generating circle. Draw radial lines from 
the center B to the pitch circle, as shown; as before stated the 
distance between points is not particular, except that what- 



J 5' *•. 3 




ever they may be in the circle K A, they must be stepped off 
the same on the generating circle C, and all must start from 
the same point A. Now with the dividers set at the center B 
describe arcs through the points 1, 2,3, 4, etc., on the gener- 
ating circle, the 10th arc passing through the 10 point on the 
generating circle to the 10th radial line ; the 9th arc passing 
through the 9th point on the generating circle to the 9th 
radial line, etc. Then with one point of the dividers set at F, 
the 10th point on the circle, and the other at G, the 10th radial 



132 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

line, transfer to the line A B at H and produce I ; and from J, 
the 9th point on the circle, to K, the 9th radial line, transfer 
to L, and produce M; continue in this manner as shown, and a 
curve drawn through these points, if not too far apart, will be 
an epicycloid for the flank of this gear. 

As already explained, in a combination of gearing in which 
they are all to interchange, the generating circle is usually 
made y% the diameter of the smallest gear in the set. But if 
you want a pair 24' ' diameter, or one that is 24" and the other 
30 /r diameter, on the pitch line, then I would use for either 
case a generating circle that was 12 r/ diameter for general 
purposes, but, if you want to increase the strength of the 
teeth, in these two latter cases I should make the generating 
circle a little less than one-half the diameter of the smallest of 
the pair, on the pitch circle. 

The teeth of racks are sometimes made with straight faces; 
some parties make them at an angle of 15° on each side, and 
some make them 14^° on each side. I prefer the latter, as 
that is the angle (29°) of a worm thread. 

BEVEL, WHEELS. 
I will now show you how to draw a pair of mitre 
wheels in gear. Figure 9 represents a pair of 6 pitch 18 teeth; 
18 -T- 6 = 3 inches for the largest pitch diameter shown at 
A A; draw these two lines first as near as possible to the 
proper length and exactly at right angles (90°) to each other ; 
next draw lines through the center of both these lines rep- 
resenting the axis of the gears and meeting at B, as shown. 
From the extreme ends of the largest pitch diameters draw 
the lines C, meeting at B. These lines (C) are the pitch lines 
of the teeth ; at right angles to the lines C B, draw E F, the 
length of which depends on the pitch of the gear; if 4 pitch, 
draw one-fourth of an inch on the outside and one-fourth plus 
the clearance on the inside. The clearance on bevel gears of all 
kinds is the same depth on the large end of the gear as in 
spur gears; as the example is 6 pitch, then draw E F one-sixth 
of an inch, or .166" on the outside of the line C shown at E 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 133 




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134 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



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THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 135 

and one-sixth plus the clearance or .192 shown at F; next 
draw all these lines B F, according to the length of the face of 
the gear wanted, toward the center B; this length and also the 
general outlines of hub, etc., depend on circumstances. The 
center angle of a pair of mitre gears is always 45°. as shown. 

It sometimes happens that we have to drive two shafts by 
gearing that are less or greater than a right angle (90°). In 
Figure 10 is an example of this kind, in which the shafts are 
at an angle of 100° from each other, as shown. The pinion 
has 16 teeth and the wheel 72, and are 8 pitch. First draw the 
lines A A and B B at an angle of 100°; at right angles to B B 
draw the line D, exactly 2 inches long (8 pitch 16 teeth) repre- 
senting the largest pitch diameter of the pinion. The end 
of this line at B should be just 4^2 inches from A A, because 
the large wheel has 72 teeth, and consequently is just 9 inches 
in diameter at the largest part (or 4% // radius). At the inter- 
section of the lines A A and B B shown at C, draw E C and G 
C; as the gears are 8 pitch, then the teeth will extend J^ // 
above the pitch line and also y^' plus the clearance below the 
pitch line at the large end only, and at right angles to the 
£itch lines B C and G C ; all these lines are drawn to the cen- 
ter at C. Draw the shape of the gears to suit circumstances. 

Figure 11 shows the manner of constructing the teeth of 
bevel gears. At right angles to the pitch line and about one- 
third of the length of the tooth from the large end, draw the 
line B B extending to the axis of the gear B B, and, with the 
dividers set to this length, transfer to one side and draw the 
circle C C. As this gear is 4 pitch, then the tooth and space 
are each .393 // wide on the pitch line ; now space off three or 
four points on the line C, and with our template or protractor 
of 75° 3(K, draw the shapes in the same manner as explained 
in spur gears, using for the circle A a radius equal to one- 
fourth of the line BB. 

Remember that the shape and dimensions of these teeth 
on the circle C C are just the same as on a spur gear of the 
same diameter; this refers only to the extreme end of the bevel 
gear tooth at the largest pitch diameter, and from this point 



136 THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 137 

the teeth gradually get smaller in all dimensions, depending 
on the length of face, etc. 

Now, we have found the shape of the teeth, but in cutting 
bevel gears the cutters will have to be made thin enough to 
cut through at the small end of the teeth, and the longer the 
face of the tooth, the thinner we will have to make our cutter. 
I have shown the shape of the cutter for this gear, and the 
arcs D D represent the pitch lines of the gear at the two ex- 
treme ends of the teeth. This will be more fully explained 
hereafter. 

I will now explain the method of finding the angles of 
bevel gears. This is a very important matter for the reason 
that if we do not know the angles, we can neither turn nor cut 
them accurately. 

Figure 12 represents a pair of bevel wheels of 4 pitch ; O 
G is a line through the center of the large wheel which has 30 
teeth, and O E, the center line through the pinion which has 
12 teeth, a portion only is shown ; the wheels are supposed to 
be in mesh, O I representing the pitch line between the two 
gears. 

Now, as the large gear has 30 teeth, we know the radius 
(or one-half the diameter) is just &%" which is radius to the 
angle BOE; we also know the radius of the pinion (or the 
length of B E), because it is one-half the diameter of the pinion 
on the pitch line, and, as the pinion has 12 teeth, dividing 12 
by 4=3 inches diameter, or lj^ 7 ' radius ; this 1J^ /7 becomes 
tangent to the angle BOE, then 1.5" divided by %-%" = 
.400", which is tangent to the radius of one inch. In the 
table of tangents we find the angle 21° 48' to correspond with 
the decimal .400. 21° 48' is then the center angle of the pinion 
on the pitch line as shown, and the difference between this 
and 90° or 68° 12' is also the center angle of the large gear, 
also shown. 

It must be remembered that the lines B O and I O are the 
pitch lines of the teeth, the latter for both the pinion and the 
large wheel as they are supposed to be in gear. 

Now, the center angle of the gear being found on the 



138 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



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pitch line, then the complement of this angle, or 68° 12 7 , will 
be the proper angle to finish the teeth A C, measured from 
the back of the gear, because A 3? should always be made 90 a 
from the pitch line B O as shown, and not from the outside or 
top of the teeth. 

The outside diameter of bevel gears is found in a differ- 
ent manner from spur gears, because in bevel gears, if of 4 
pitch, as in the example, we measure J4" (B A) at right angles 
to the pitch line B O. 

For convenience I have shaded a small angle A B D at the 
top of this pinion, and, in order to show it more clearly have 
removed it to one side and have also enlarged it. This angle is 
for the purpose of finding the exact diameter of this pinion 
at the highest point shown at A. 

In the angle A B D, the longest side A B is known, be- 
cause it is always equal to the pitch. Thus in 10 pitch, it 
would be .100"; in 8 pitch, .125 // , and in 4 pitch, .250". As 
the example is 4 pitch, then A B is .250". We also know the 
angle, because it is always the same as the center angle on the 
pitch line; thus, in the figure the angle is 21° 48', and A D is 
sine and D B cosine to this angle. Now, in the table we find 
the cosine of 21° 48' = .928, and, multiplied by the radius 
(.250) = .232", and this is the distance D B to be added to the 
radius of the pinion; that is, the radius is 1.5"; then 1.5" + 
.232=1.732" for one side, and doubled = 3.4642" for the 
largest diameter of the pinion. 

Let us now find the angle to turn the face of this pinion. 
We have found the center angle at the pitch line, and for con- 
venience we now want to know the length of this pitch line, 
(B O). As O E is radius and B K tangent, then B O is the 
secant of the angle BOB. In the table of secants we find 
opposite 21° 48' the decimal 1.077, which, multiplied by the 
radius 3% = 4.0387, or say 4.039" for the length O B, as 
shown. Now as A C is always at right angles to B O, then we 
have a right angled triangle A O B, figures 12 and 13. In the 
latter figure I have separated that portion of the tooth above 
the pitch line, and also that portion including the clearance 



140 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 






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THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 141 

below the pitch line, forming two separate right angled trian- 
gles : the one above the pitch line for finding the angle to turn 
the pinion, and the one below the pitch line for finding the 
angle to cut the teeth. 

As A C is 90° from O B and the length is also known to 
be 4.039", also that A O is longer than B O, then A O is the 
secant to the angle A O B. 

We know A B is .250", and divided by the radius 4.039 = 
.0618+, or the tangent to the radius 1, and in the table of tan- 
gents this corresponds to the angle 3° 32', which added to 21° 
48' = 25° 20' for center angle; or 90° — 25° 20' = 64° 40' 
measured from the back as shown, for turning the face of the 
pinion. 

For cutting the teeth the angle B O C is shown, and 
we have to add the clearance, which is always on the large end 
of bevel wheels the same as in spur wheels; in 4 pitch the clear- 
ance is .039", and added to .250" = .289", which divided by 
4.039 = .0715 4- = tangent to radius 1, which corresponds to 
the angle 4° 6', and 21° 48' — 4° 6' = 17° 42', for the center 
angle; or 90° — 17° 42' = 72° 18', the angle to cut this gear, 
measured from the back as shown in Figure 12. 

It is also necessary in making cutters for bevel gears to 
know the smallest pitch diameter of the gears in order that 
the cutters may be made of the proper thickness on the pitch 
Line for cutting them. 

Figure 14 shows the method of finding this diameter 
[figures 12, 13 and 14 representing the same gear.) 

I have shown that the line B O is 4.039", and in figure 12 
he face is marked 1^"; then 4.039 — 1.25" = 2.790" nearly. 

In figure 14 I have shaded an angle A O C for the purpose 
>f showing the method of finding the smallest pitch diameter 
>f any mitre or bevel gear. 

We have found the center angle on the cone pitch line to 
>e 21° 48', and as the longest line (A O) is radius, then A C 
becomes sine, and the sine of 21° 48' = .371 + , which multi- 
plied by the radius (2.790") = 1.036" for the length of A C, 



142 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 







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THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 143 

and twice this = 2.072" for the smallest pitch diameter of the 
pinion. 

2.072" X 3.1416 = 6.509" for the circumference, and, 
divided by twice the number of teeth (24) = .2712", for the 
thickness of the cutter or tooth on the pitch line. 

It must not be forgotten that the cutter is made of the 
proper thickness to suit the teeth on the smallest pitch diame- 
ter, and, in making the cutter, it is also necessary to know the 
depth from the pitch line to the bottom of the tooth on the 
small end. 

Referring back to Figure 14, I show a method of finding 
this depth by the angle EOF. I have shown A O to be 
2.790" long, and this line is represented by E O. Now this 
angle is already explained in Figure 13, which is the same 
thing, and has been found to be 4° 8 7 ; then as E O represents 
the pitch line, and E F at right angles to it the tangent, we 
multiply the tangent of 4° 6' by the radius 2.79; thus tangent 
4° 6' = .0717 X 2.79 = .200" for the depth from the pitch 
line to the bottom of the teeth on the small end of the gear. 

Now, as the teeth of bevel gears are of the same depth, 
thickness, etc., at the largest pitch diameter as those of spur 
gears, then this gear of 4 pitch will be .289" from the pitch 
line to the bottom of the teeth on the large end, and only 
.200" on the small end, a difference of .089" in the depth from 
the pitch line, and of course, the cutter will be too thin for the 
large ends of the teeth. 

To finish the teeth of bevel wheels correctly (or I might 
say approximately correct, for bevel gears cannot be made 
correct by rotary cutters) we will have to pass the cutter 
through each space at least twice. 

There are two ways of cutting these teeth to get them of 
the proper thickness on the large end of the teeth, and, if pos- 
sible, the wheels should always be on an arbor that is fitted in 
the taper hole of the index head spindle, no matter how slight 
the angle of the teeth may be, as otherwise the teeth may be 
thicker on one side than the other (or rather not spaced uni- 
ormly). 



144 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 

One way of cutting the teeth of bevel wheels to get them 
the right thickness on the large end of the teeth is to place 
the cutter central with the headstock spindle, and, having the 
wheel elevated to the proper angle, pass the cutter through, 
then rotate the wheel by the index crank, say two or three holes 
on the index plate, depending altogether on the gear you are 
cutting, that is, whether the cutter is much thinner than 
standard size. 

Now, if you revolve the wheel as stated, but one hole, you 
will have to move the table slightly in or out depending upon 
which way you revolve it, so that the cutter will pass through 
the small end of the gear without touching the sides of the 
tooth, and, as the large end of the wheel will revolve through 
a greater space than the small end, so will the cutter passing 
through take out more and more as it reaches the large end 
of the teeth. 

Remember that whatever the number of holes you turn 
one way, you must also turn the same number in the opposite 
direction, otherwise you will get crooked teeth. 

It can easily be seen that the teeth will be rounded off a 
little too much unless the cutters have been made for cutting 
in this way. 

The other way for cutting the teeth of bevel wheels is to 
set the Universal Head at the proper angle to the table, and 
cutting all the teeth in this position, then swivel the head to 
the same angle in the opposite , direction and pass the cutter 
through the second time. The greatest care should be used 
to have the cutter central with the work. 

I will now show you how to get the exact angle for swiv- 
eling the Universal Head to cut the teeth of a bevel wheel. 

In figure 15, which also corresponds with Figure II, but 
greatly enlarged, we have found the thickness of the tooth, or 
space at the pitch line on the small end of the gear to be .247 "\ 
we will also have to know the thickness of the cutter that 
touches the pitch line as it passes through the large end. 

In figure 11 I have shown the position of the template of 
75° 30' on the pitch line, with the circle A equal to one-fourth 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



145 




146 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 

04) the radius, thus 39/16" -r- 4 = .890". Now in Figure 15 
the pitch line would touch at A 1 , which corresponds with C C, 
Figure 11. In Figure 15 H H represents the template of 75° 
30' in line with the center of the gear and a right angled tri- 
angle (A 1 O representing the top of the template) with the ver- 
tex at O as shown, will be the complement of 75° 30' or 14° 
30' as shown in the figure. We have found the depth from 
the pitch line to the bottom of the tooth on the small end of 
the gear, which is .182", and we know that the depth from the 
pitch line to the bottom of the tooth on the large end is the 
difference between the whole depth .539" and .250", or .289" 
(for 4 pitch); then draw two lines D D on this cutter as shown, 
which represents the height of the pitch lines at the two ends 
of the teeth in the bevel gear. 

We know how wide the space should be on the pitch line 
at the large end of the tooth, because it is the same as any 
tooth of a 4 pitch spur gear or .393"; as the cutter was made 
thin enough for the small end, we will have to find the width 
across the cutter that touches at the largest pitch diameter by 
the angles shown. In other words, the object now is to find 
the distance between 1 1 and H H. As A 1 O is .890" we call 
it radius, A 1 B the sine and B O cosine of the angle A 1 O B. 

The cosine of 14° 30' = .968 + and multiplied by the 
radius (.890") = .8626" for the length of B O. Now we have 
found the depth from the pitch line to the bottom of the small 
end of the gear (or the cutter) to be .182" as stated, and the 
depth on the large end .289"; then the difference between 
these pitch lines, as represented by the lines D D, Figure 15 = 
.107"; we next find the distance A 1 B, or the sine of the angle 
A 1 O B. The sine of 14° 30' = .2503 and multiplied by the 
radius (.890) = .223"; then subtracting the distance between 
D D which we said was .107" from .223", we have .116" for 
the sine C G of the second angle (the line C O being dotted). 

Now to find what this angle is we divide .116" by .890 and 
we have .1301 for the sine to a radius of 1 inch, which in the 
table corresponds to the angle 7° 29 / . 

We have now found the degrees and minutes of these two 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 147 

right angled triangles, and we now want to know the cosines 
of these two angles in order to know the distance between the 
two lines 1 1 and H H. The cosine of 14° 30' = .968, and multi- 
plied by the radius (.890) = .861". The cosine of 7° 29 / = 
.9915, and multiplied by the radius (.890) = .883. Now the 
difference between .861" and .883" = .022", and this is the 
distance between 1 1 and H H. 

Now draw an angle as shown dotted at the bottom of fig. 
ure 15, representing the length of the teeth (which in this case 
is one inch), one end equal in width to the thickness of 
the space between two teeth at the small end on the pitch 
line, the other end equal to the width of the space be- 
tween two teeth on the large end on the pitch line, minus 
twice the width of space between the lines I I and H H ; this 
will be the correct angle for cutting the teeth of this bevel 
gear. 

The angle for cutting any bevel gear or mitre wheel can 
also be found in a similar manner. 

SPIRAL GEARS. 

Figure 16 shows the manner of finding the correct diame- 
ter of gears when they are to be cut spiral and with regular 
standard cutters. Find the diameter oi the gear on the pitch 
line in the usual manner and multiply it by the secant of the 
angle you wish to cut it, then add two pitches in the usual 
way. Thus, suppose you want a spiral gear of 8 pitch 24 
teeth, or 10 pitch 30 teeth. The diameters for regular spur 
gears would be 3" diameter on the pitch line. But if you 
want them cut at an angle of 10°, then multiply the secant of 
10° = 1.0154 by 3", and the result, 3.046", will be the correct 
diameter on the pitch line; if 8 pitch, add two 8ths to this, 
making 3.296" for the diameter of the outside. If the same 
gear was to be 10 pitch, then add two lOths, making it 3.246" 
outside diameter. 

Now, if the same gear was to be cut at an angle of 23° 10', 
then the secant of 23° 10' = 1.0877, multiplied by 3" =3.263" 
for the diameter on the pitch line. 



148 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



149 



If they were to be cut at an angle of 45°, then the 
secant of 45° = 1.414 X 3" = 4.242" for the diameter 
at the pitch line. 

Suppose you want a spiral gear of 4 pitch, 36 teeth 
and 10° angle. 36 -H 4= 9, this multiplied by the secant 
of 10° = 9.1386" for the pitch diameter and 2/4ths 
added = 9.6386 r/ for the outside diameter of the gear. 

It sometimes happens, however, that we have a 
standard size blank, say 2%" diameter on the pitch 
line, and we want it to have 22 teeth ; this would be 
right for an ordinary spur gear of 8 pitch. But if we 
should want to cut this gear at an angle of 45° (spiral), 
then we would have to make a special cutter, 



<6* 



A3 



/ 



2,2.751 -y 






150 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

unless it would suit some ether pitch. To find the thick- 
ness of the cutter on the pitch line when the diameter 
of the gear (on the pitch line) and the number of teeth is 
given, multiply the cosine of the angle you wish to cut it by 
the diameter of the gear on the pitch line, and this again by 
3.1416, then divide the product by twice the number of teeth, 
and the result will be the thickness of the tooth on the pitch 
line, thus : 

In the example, figure 17, suppose we have a gear blank 
2%f / diameter on the pitch line, and we want the teeth to be 
cut spiral at an angle of 45°, 22 teeth. The cosine of 45° = 
.707, and .707 X 2.75" = 1.9442", and this again multiplied 
by 3.1416 = 6.105"; then 6.105" -r- 44 = .138 + " for the thick- 
ness of the cutter on the pitch line. This would be nearly 
right for a cutter of 11 pitch, and, when the difference is so 
slight, we can sometimes make the gear a little larger or 
smaller to suit circumstances. Thus, suppose instead of mak- 
ing it 11 pitch, which is a little thicker than in the example, 
we make the blank to suit a 12 pitch cutter, which is about 
.007" thinner, then, as there are 22 teeth and 22 spaces, a 12 
pitch cutter being .131" thick on the pitch line, we multiply 
.131 by 44 = 5.764"; divide this by 3.1416 = 1.834" which 
multiplied by the secant of 45° (1.414) = 2.593", or the diame- 
ter. 

Suppose a party came into the shop and ordered a pair of 
spiral gears, one of the gears to be twice as large as the other 
to connect or drive two spindles that were 6" apart, and 
parallel, and the teeth to be cut at an angle of 10° from their 
axis. What will be the thickness of the cutters at the pitch 
line? 

If the spindles are 6" apart, and one twice as large as the 
other, then the radius of one gear will be 2" and the other 
4", or the diameters 4" and 8". The cosine of 10° = .9848, 
and multiplied by 8" (the large wheel) = 7.878", and this 
again multiplied by 3.1416 = 24.738"; this divided by twice 
the number of teeth in the wheel will be the thickness of the 
cutter on i he pitch line. Thus: suppose therj are 48 teeth 



THE MACHINIST AND TOOT, MAKEE's INSTRUCTOR. 151 

on the wheel, then 24.738 -7- 96 = .257 + ", the answer; and 
this, of course, will be the thickness of cutter for the small 
wheel also. 

Now as these shafts are to run parallel, then one of the 
wheels will be cut right hand and the other left hand. They , of 
course, will be of the same angle, 10°, but the pitch of the large 
wheel will be twice as great as the small one. 

In cutting the teeth of spiral gears the angle is usually 
given, but sometimes, instead of the angle, the number of 
inches to one turn is given ; thus, suppose you have a spindle 
2J£" diameter, and you want to mill grooves on it that will 
have, say one turn in 30". In doing work like this, it is nec- 
essary that we know the proper angle to swing the table, which 
is done as follows : Multiply the diameter by 3.1416 and divide 
the product by the number of inches to one turn of the groove, 
which will give the tangent. In the table of tangents the 
angle can be found. Thus, 2.5X3.1416 = 7.850; 7.850 -r- 
30 = .2617, the tangent, which, in the table corresponds to 
the angle 14° 40', the answer. 

In cutting spiral gears, standard cutters (as regards shape) 
can generally be used, excepting when the angle is very great 
and the wheel is very small. In the latter case the teeth will 
be rounded off too much. If the cutter is large in the diame- 
ter at the same time, it will also increase the trouble. Special 
cutters should be made for this class of work if accuracy is 
required. 

It must be remembered that in milling most kinds of 
work when the angle is very great, that the shape of the 
groove will not be the same as the cutter that produced it, 
except when using end mills. In this case the table is not 
set over, but end mills, if small, are too slow and consequently 
expensive. 

WORM GEARING. 

In making worms and worm wheels it is more convenient 
to do so by circular pitch than diametral pitch, for the reason 
that a worm is nothing more or less than a screw with a special 



152 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

shape of thread, (29° angle, 14^° each side). As all lathes 
are geared to cut so many threads per inch, for instance 4 
threads per inch, which means X 7/ pitch, then the worm 
wheel must also be cut to suit it, and for this reason we use 
circular pitch for this class of work. 

If you want to cut a worm and worm wheel with 9 threads 
per inch, dividing 1 by 9, the result .111 + " would be the pitch 
of both the worm and worm wheel, and dividing by 2, the re- 
sult, .0555 // , will be the thickness of the thread and also the 
tooth on the pitch line. 

Most parties in ordering worms and worm wheels will say 
3 pitch, 4 pitch, 5 pitch, etc., but they simply mean so many 
threads per inch (called linear pitch), and means so many 
threads measured in a straight line. 

Figure 18 represents a worm and worm wheel of %" cir- 
cular pitch; the wheel has 84 teeth. When the distance be- 
tween centers should be some even figure, say 6", 8", etc., we 
make the worm wheel the right diameter to suit the pitch and 
number of teeth required, and then the worm is made of the 
proper size to " fill out," as you would call it. 

In the figure the distance between centers is not import- 
ant, and I have shown the worm to be 4" outside diameter. 

As the wheel has 84 teeth and is %" pitch, we multiply 
% or .375 by 84, and we have 31.5 inches for the circumference, 
which, divided by 3.1416 = 10.0267 inches for the diameter on 
the pitch line, as shown at K ; we find the working depth for 
%" pitch to be . 238", and one-half of this or .119" added to 
each side will make the worm wheel 10.265" diameter at the 
throat, as shown on the line H. This is the only thing about 
worm wheels of much importance, that is, to get the correct 
diameter at the throat. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 153 




154 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 





WORMS AND WORM WHEELS. 






a W 




THICK- 






WIDTH 




CIRCU 


« z 


DIAME- 


NESS OF 


WORKING 


WHOLE 


OF 


WIDTH OF 


LAK 


a S 


TRAL 


TOOTH ON 


DEPTH OF 


DEPTH OF 


THREAD 


THREAD 


PITCH. 




PITCH. 


PITCH 
LINE. 


TOOTH. 


TOOTH. 


TOOL AT 
END. 


AT TOP. 


8" 


J6 


1.0472 


1.5000 


1.9098 


2.0598 


.9285 


1.0061 


m 


4/11 


1.1423 


1.375 


1.7506 


1.8881 


.8511 


.9223 


w* 


2/5 


1.2566 


1.250 


1.5915 


1.7165 


.7737 


.8384 


2M 


4/9 


1.3962 


1.125 


1.4324 


1.5449 


.6963 


.7545 


2 


K 


1.5708 


1.0 


1.2732 


1.3732 


.6190 


.6707. 


m 


8/15 


1.6755 


.9375 


1.1936 


1.2874 


.5803 


.6288 


i% 


4/7 


1.7952 


.875 


1.1141 


1.2016 


.5504 


.5869 


m 


8/13 


1.9332 


.8125 


1.0345 


1.1158 


.5029 


.5449 


iy 2 


% 


2.0944 


.750 


.9549 


1.0299 


.4642 


.5030 


Ws 


8/11 


2.2847 


.6875 


.8754 


.9441 


.4255 


.4611 


m 


4/5 


2.5132 


.625 


.7958 


.8583 


.3870 


.4192 


V/8 


8/9 


2.7925 


.5625 


.7162 


.7724 


.3482 


.3770 


1 


1 


3.1416 


.500 


.6366 


.6866 


.3095 


.3353 


15/16 


11/15 


3.3510 


.4687 


.5968 


.6437 


.2902 


.3144 


% 


11/7 


3.5903 


.4375 


.5570 


.6008 


.2710 


.2934 


13/16 


13/13 


3.8664 


.4062 


.5173 


.5579 


.2514 


.2724 


3/4 


i>6 


4.1888 


.375 


.4774 


.5149 


.2321 


.2515 


11/16 


15/11 


4.5694 


.3437 


.4377 


.4720 


.2130 


.2305 


X 


13/5 


5.0264 


.3125 


o3978 


.4290 


.1935 


.2096 


9/16 


17/9 


5.5850 


.2812 


,3581 


.3862 


.1741 


.1886 


% 


2 


6.2831 


.250 


.3183 


.3433 


.1550 


.1676 


7/16 


2 2/7 


7.1806 


.2187 


.2788 


.3006 


.1353 


.1466 


2/5 


2^ 


7.8540 


.200 


.2546 


.2746 


.1242 


.1342 


% 


2% 


8.3776 


.1875 


.2387 


.2575 


.1160 


.1257 


Ks 


3 


9.4248 


.1666 


.2122 


.2289 


.1032 


.1116 


5/16 


3 1/5 


10.0528 


.1562 


.1989 


.2145 


.0967 


.1048 


2/7 


3^ 


10.9955 


.1429 


.1819 


.1962 


.0885 


.0956 


M 


4 


12.5663 


.125 


.1591 


.1716 


.0799 


.0838 


2/9 


43^ 


14.1371 


.1111 


.1415 


.1526 


.0686 


.0741 


1/5 


5 


15.7079 


.100 


.1273 


.1372 


.0620 


.0670 


3/16 


5% 


16.7552 


.0937 


.1193 


.1287 


.0580 


.0628 


1/6 


6 


18.8496 


.0833 


.1061 


.1144 


.0515 


.0556 


1/7 


7 


21.9912 


.0714 


.0910 


.0981 


.0442 


.0478 


1/8 


8 


25.1327 


.0625 


.0795 


.0858 


.0386 


.0419 


1/16 


16 


50.2654 


.0312 


.0398 


,0429 


.0193 


.0209 



This table refers to single threads only. 
If double threads are required divide by 2. 
If triple by 3, and if quadruple by 4, etc. 



THE MACHINIST AND TOOE MAKER' S INSTRUCTOR. 155 

Now, if it was necessary that the distance between centers 
should be say 6", then we would take one-half of the diameter 
or the radius of the worm wheel, as shown on the pitch line 
at G = 5.0133", and subtracting it from 6" we would have 
.9867" for one-half the diameter of the worm on the pitch line, 
or 1.973" for the whole diameter on the pitch line, and adding 
.238" we would have 2.211" for the outside diameter of the 
worm. 

The exact diameter of any worm wheel at the largest part 
as shown on the line I, although not very important, can easily 
be found if we know the angle of the sides of the teeth; in the 
figure the angle is 30°, as shown. 

For convenience I have shown a right angled triangle at 
the center of the worm A O B; the length O A is known, be- 
cause it is one-half of the diameter of the worm, less the work- 
ing depth (.238") of the tooth ; then 2" — .238 + " = 1.761" 
for the radius A O ; the cosine of 30° = .866, which multiplied 
by the radius 1.761 = 1.525", for the distance O B. 

Now we know that from the center of the worm to the 
throat of the worm wheel it is 2" — .238", or 1.761"; then 
subtracting 1.525" from 1.761" = .236" in a vertical direction 
from B to the throat of the wheel at the center ; then twice 
this, or .472", added to 10.265" = 10.737" for the largest 
diameter of the worm wheel, as shown on the line I. 

The dimensions of worm threads and the width of tools 
for cutting them are found in the following manner: 

In the example the thread is %" circular pitch, and divid- 
ing 3.1416 by %, or .375, we have 8.378 for the diametral 
pitch; then dividing 1 by 8.378 = .1193" for one half of the 
working depth, or .238", for the working depth of the tooth or 
thread ; as the pitch is %" ', then the tooth and also the space 
is one-half or 3/16" thick on the pitch line ; one-tenth of 3/16", 
or .187" = about .019", and this added to twice .119" = .257", 
for the whole depth of any worm, worm wheel, or spur wheel 
of %" circular pitch. 

The width of the top of thread and also the bottom of the 
space is found in the following manner : 



156 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



Figure 18% represents the side of this %" pitch tooth 
greatly enlarged, shown at A A. The angle of worm threads, as 
explained, is 14%° on each side, and drawing a line perpen- 
dicular to the pitch 
line, as shown, 
we produce a right 
angled triangle on 
each side of the 
pitch line, with the 
radius given. Then, 
as the tangent of 
14%° = .2586, we 
multiply this by 
.119", and the re- 
sult, .0307", is the 
distance B B, and 
twice this, or .0615, 
subtracted from 
the thickness of 
the tooth on the 
pitch line (.1875)" 
= .125" for the 
width of thread on 
top. The tangent, .2586, multiplied by .138", the distance 
from the pitch line to the bottom of the tooth, thus, .2586 X 
.138" = .0356 for the distance C C; twice this, or .0713", 
subtracted from the thickness of the tooth on the pitch line = 
.116", nearly, for the width of space on the bottom, or the 
width of the tool for cutting the thread, etc., for a worm of 
%" circular pitch, 

The dimensions of any worm thread can be found in the 
same manner. 

In cutting the teeth of worm wheels they should first be 
driven on an arbor, and the table set over to the proper angle 
(corresponding to the angle of the worm thread), then select 
an ordinary spur gear cutter, not much, if any, larger in the 
diameter than the worm, and a little thinner on the tooth than 




THE MACHINIST AND TOOl, MAKER' S INSTRUCTOR. 157 

the thread of the worm, and placing the worm wheel directly 
under the center of the cutter, raise the wheel up into the 
cutter sufficiently to cut it two-thirds or more of the whole 
depth, depending upon the size and shape of cutter, and also 
whether the angle is correct. The teeth are roughed out only 
in this way by indexing; after which a hob, made like the 
worm, and with a diameter equal to that of the worm, plus 
twice the clearance added, is placed on the arbor of the milling 
machine, with the table set back to zero (right angled to the 
cutter arbor) ; the dog is then taken off the worm wheel arbor, 
and the wheel is then raised up into the hob as before. The 
hob will revolve the worm wheel while it is gradually raised to 
the proper depth. 

Now let us see what the angle should be in this worm 
wheel. As the worm is 3.761" diameter on the pitch line, 
then the circumference would be about 11.809", and dividing 
the pitch by the circumference, thus, %" = .375 -r- 11.809" = 
.0317" for the tangent (this will be more fully explained here- 
after). This tangent corresponds to the angle 1° 49", which 
is the correct angle to swivel the table for roughing out the 
worm wheel. 

Remember to take the circumference on the pitch diame- 
ter of the worm, and not the outside diameter. The angle for 
roughing out any worm wheel is found in a similar manner. 

It sometimes happens that when worm wheels are very 
small, or of a very coarse pitch, that it is necessary when com- 
mencing to hob the wheel (after it has been roughed out) to 
help revolve it by hand until it has made two or three turns, 
after which it will take care of itself. 

In figure 19 I have shown an attachment to be 'used on the 
Universal Milling Machine for cutting the teeth of worm 
wheels complete, by hobbing, when a great many of one size 
are wanted, the details of which are as follows: A A is a 
spindle fitted at one end (shown at the left) to the milling 
machine spindle, the outer end projecting through the bear- 
ing jB, which is held sidewise by the* bevel wheel C on one 
side and a tight fitting nut and collars on the other, as shown. 



158 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 159 

The outer end of this spindle is also bored taper to receive the 
shanks of small hobs, the outer end of the hob being supported 
by the center in the overhanging arm. The bevel wheel C 
meshes with D, which is screwed on the shaft B. The bear- 
ing F has a hollow shaft or sleeve passing through it, and is 
bored to slip over the worm shaft on the Universal Head, 
which it is to drive. The mitre wheel G is supposed to work 
on either side of the bearing F, in order to cut both right hand 
and left hand worm wheels as desired. 

In the figure I have shown 40 teeth on the driving wheel C, 
and 30 teeth on the driven D. The pair at the other end are 
mitres, and can be made any size to suit circumstances (say 30 
teeth each). As it takes 40 turns of the worm shaft to make 
one revolution of the spindle in the Universal Head, then, if 
we want to cut a worm wheel of 30 teeth, a pair of bevel 
wheels with 40 teeth for the driver, and 30 for the driven, as 
shown, will give the required number of teeth. If 32 teeth 
are wanted on the worm wheel, then a bevel wheel with 32 
teeth should be placed at D, and if 50 teeth are wanted, then 
the wheel at D should have 50 teeth also. Of course, the 
other wheels need not be changed as regards the number of 
teeth. 

The worm wheel to be cut is placed on an arbor between 
the centers in the usual way, and driven by a dog, the attach- 
ment being made long enough to suit the work. As the shaft 
H telescopes the shaft E, we are not limited to an exact 
length of the arbor on which the work is placed. Both bear- 
ings F andB are of the same design and better understood by 
the side elevation shown at I, partly broken off at one of the 
bearings for want of space. 



160 THE MACHINIST AND TOQIL MAKER'S INSTRUCTOR. 



CHAPTER III. 



Extracts from Brown & Sharps Manufacturing Co.'s 

Treatise on Milling Machines. 

(published by permission.) 



MHJJNG MACHINES AND SKIU,. 

No one who has had sufficient experience with the Milling 
Machine in its various forms to acquire a reasonably clear idea 
of its capabilities, and who has an opportunity to see the ma- 
chine in use in the various shops, can fail to see that in many 
of them it is very imperfectly understood, and that, as a con- 
sequence, comparatively poor results are obtained from its 
use — results, we mean, which are very poor compared with 
those which should be obtained, and are obtained in every 
case where the legitimate functions of the machine are clearly 
recognized, and the conditions necessary to its successful 
operation secured. 

The Milling Machine intelligently selected or constructed, 
with reference to the work it is expected to do, provided with 
well designed and well made special fixtures, where the nature 
of the work calls for them, and then skillfully handled, is a 
surprisingly efficient tool, but used as it is being used in many 
shops today, it is a delusion, a failure and an injury alike to 
the users, to the builders and to the good name of the mill- 
ing machines generally. 

While it is true that there is scarcely a machine tool in 
use which will yield more satisfactory returns for a given out- 
lay, when pains are taken to use it in the best possible man- 
ner, it is also true, we think, that there is no tool in common 
use, the efficiency of which is so much reduced by careless or 
ignorant handling and abuse. Considerable intelligence and 
skill, as well as constant attention, must be bestowed upon 
the milling machines in order to secure anything like a satis- 
factory performance from them, either in the quality of the 
work done or in its quantity. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 161 

In some cases this skill and intelligence must be pos- 
sessed and exercised by the man who actually handles the 
machine; in other cases by some one who, though he does not 
actually operate it, supervises its operation, and is responsible 
for the work done by it. But in any case, the skill, the intelli- 
gence and the careful attention must be exercised, or the re- 
sults will be anything but satisfactory. 

We hear a great deal about the comparatively cheap labor 
required to do milling machine work, and it is evident that 
too many shop proprietors have concluded from this that 
about all that is necessary to do such work is to buy the ma- 
chine, hire a boy to run it, have him shown how for an hour 
or so by one of the lathe hands, and then let the boy and the 
machine work out their own salvation. 

No greater mistake could possibly be made, and it is in such 
a shop that a milling machine man findc: the machine working 
often at less than half of its capacity, with an apology for a cut- 
ter, ground by hand in every shape but the right one, two or 
three only of its superabundant ieeth touching the work, and 
they, with a distinct thump and knock, indicating anything 
but a real cutting action, while the boy stands by, and occa- 
sionally — when it occurs to him to do so — squirting a few 
drops of black lubricating oil onto the chips with which the 
spaces between the teeth are tightly jammed. The proprietors 
of such shops are not usually very enthusiastic regarding the 
use of milling machines, and it would be a wonder if they 
were. 

Where a Universal Milling Machine is used upon tool 
work or for other purposes requiring a constant change from 
one job to another, it is a mistake to suppose that there is 
economy in the employment of a boy or cheap man to operate 
it. Many of those who think they are saving money in 
that way would be greatly surprised to see the work turned 
out from such machines by good mechanics who thoroughly 
understand them, and are capable of earning good wages 
upon them. 

Experience has proved that it pays as well to put first- 



162 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 



class mechanics upon such machines as upon any other ma- 
chine tools. 

Where milling machines are used for regular manufactur- 
ing operations, and the same cycle of movements is to be 
repeated for a large number of pieces, boys or men who are 
not skilled mechanics, answer every purpose ; but the skilled 
supervision must be there, and it must be seen to that the 
machine is as well taken care of, the cutters as well made and 
ground, and in fact, everything as well done as though a good 
mechanic actually operated the machines. In fact, in the 
shops in which the best results are obtained from the use of 
milling machines in regular manufacturing operations, all 
changes of the machines from one job to another, all adjust- 
ments, and the grinding and replacing of cutters, are done by, 
or at least under the direct supervision of, a skilled mechanic, 
responsible for the work of the machines, and who thoroughly 
understands and appreciates them. In this way only can the 
full benefits of the machine be realized. 

It is far too common to go into the tool room and find a 
splendid Universal Milling Machine standing idle, while per- 
haps two or three men are doing at the shaper,planer or vise, jobs 
which could be done by an expert milling machine hand in one- 
fourth down to one-tenth of the time and a great deal better. 
One fault, which is far more common than would readily be 
believed in some quarters, is a failure to recognize the fact that 
a milling machine necessarily calls for some sort of a machine 
for grinding cutters, and that a machine upon which cutters 
are used, that are ground by holding the edges one after the 
other against an emery wheel by hand, is at a decided disad- 
vantage, and will do no work which either in quality or cost 
will make a favorable showing when compared with that which 
is done by properly ground cutters. 

It should be much more generally recognized that in mill- 
ing machine practice, as in other things, there is a right way 
and a wrong way, and that skilled, intelligent labor pays best. 
When these facts are more generally recognized, it will be 
better for both the builders and for the users of the machines. 



THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. 



1G3 




For most purposes, it is well to have mills or cutters as small 
in diameter as the work or their strength will admit. The 
reason is shown by figure 41. Suppose the piece IDCJE 
is to be cut from I J to D B. If the large mill A is used, it will 



164 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. 

strike the piece first at I, when its center is at K, and will 
finish its cnt when its center is at M. The line G shows how 
far the mill must travel to cut off the stock I J D K. If the 
small mill B is used, however, it travels only the length of the 
line H. It can also be seen that a tooth of B travels through 
a shorter distance between the lines D E) and I J than a tooth 
of A. This is true of all ordinary work, or where the depth of 
cut I D is not more than half the diameter of the small mill. 

The advantage of small mills has been illustrated in our 
own works where a difference of y 2 an inch in the mills has 
made a difference of 10 % in the cost of the work. 

In short, small mills do more and better work, cut more 
easily, keep sharp longer and cost less than large mills. 

When it is possible the mill should be wider than the 
work, and the hole in the mill should be as small as the 
strength of the arbor will admit. The stock around the hole, 
however, should not be less than % of an inch thick. 

A mill is not necessarily too soft because it can be scratched 
with a file, for sometimes when cutters are too hard, or 
brittle, and trouble is caused by pieces breaking out of the 
teeth, they can be made to stand well and do good work by 
starting the temper. 

Of late years mills have been made with coarser teeth than 
formerly; the advantages being more room for the chips and 
less friction between the teeth and the work. When the teeth 
are so fine that the mill drags, or the stock is powdered, the 
mill heats quickly and does not cut freely. 

The friction may also be reduced, especially in large mills 
taking heavy cuts, by nicking or cutting away parts of the 
teeth, which breaks the chips and allows heavier cuts and feeds 
to be taken. 

Knowing the conditions under which a mill is to be used, 
in our own practice, we modify the number of teeth as seems 
expedient, usually making the special mills coarser in pitch 
than the stock mills, for our observation indicates that there 
are more mills with too many teeth than with too few. But 
sometimes we relatively increase the number of teeth, as for 



THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 165 

instance, large mills, in some cases, can advantageously be 
designed to have more than one tooth cutting all the time on 
broad surfaces and in deep cuts. 

The adoption of the most suitable cutting angle should 
receive the same close attention that is now universally be- 
stowed upon the ordinary tools for turning and plan- 
ing. But in practice, while in many instances adopting the 
angle according to the material to be used, yet taking into con- 
sideration all the conditions of using and caring for the 
cutters, we have generally found it satisfactory to make the 
cutting edges of the teeth radial. 

The relief or clearance of mills we think should usually be 
about three degrees, and the land at the top of the teeth from 
.02 inches to .04 inches wide before the clearance is cut or 
ground. 

Mills to cut grooves should be hollowed about five one- 
thousandths in one inch for clearance; that is, a grooving mill 
should be about one one-hundreth of an inch thinner at one 
inch from its edge or circumference than it is at the edge. 

Our grooving mills are given a limit of two one-thous- 
andths in thickness. Mills made to exact thickness are very 
' expensive. In cutting grooves that are to have some parts 
of their sides nearly or quite parallel, it is well to leave con- 
siderable stock for the finishing cut, as mills, like taps, do 
better work when they get well into the stock. 

A dull mill wears away rapidly and does poor work. Ac- 
cordingly care must be taken to keep mills sharp. In sharp- 
ening them it is necessary to be very careful that the temper 
should not be drawn. The emery wheel should be of the 
proper grade as to hardness and as to the size of the emery. 
The wheel should be soft enough so that it can be easily 
scratched with a pocket-knife blade, and the emery should 
not be finer than 90, nor coarser than 60. As a rule, the 
coarser and softer the wheel, the faster it should run, although 
the periphery speed should not exceed 5,000 feet per minute. 
A wheel of the proper grade should be used with the face, not 
to exceed %" wide. If the wheel glazes, the temper of the 



166 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

cutter will be drawn. In such a case, if the wheel is not 
altogether too hard, it can sometimes be remedied b}' reducing 
the face of the wheel to about Jg", or by reducing the speed, 
or by both. 

Before using, a wheel should be turned off so that it will 
run true. A wheel that glazes immediately after it has been 
turned off, can sometimes be corrected by loosening the nut 
and allowing the wheel to assume a slightly different posi- 
tion, when it is again tightened. 

Another method of preventing a wheel's glazing is to use 
a piece of emery wheel, a few grades harder than the wheel in 
use, on the face of the wheel, whereby the cutting surface of 
the wheel is made more open and less apt to glaze. 

Mills that have their teeth ground for clearance are par- 
ticularly apt to have their temper drawn in sharpening, espe- 
cially at the edge of the teeth, and often when the temper has 
been drawn and the teeth are polished, they will look as usual 
after being ground. 

Oil is used in milling to obtain smoother work, to make 
the cutters last longer, and where the nature of the work re- 
quires, to wash the chips from the work or from the teeth of 
the cutters. It is generally used in milling a large number of 
pieces of steel, wrought iron, malleable iron ? or tough bronze. 
When only a few pieces are to be milled it frequently is not 
used, and some steel castings are milled without oil; also in cut- 
ting cast iron it is not used. For light, flat cuts it is put on the cut- 
ter with a brush, giving the work a thin covering like varnish; 
for heavy cuts it should be led to the cutter from the drip can 
sent with each machine, or it should be pumped upon or across 
the cutter in cutting deep grooves ; in milling several grooves 
atone time, or indeed, in milling any work, where, if the 
chips should stick, they might catch between the teeth and 
sides of the groove and scratch or bend the work. 

Generally we use lard oil in milling, but any animal or 
fish oils may be used. The oil may be separated from the 
chips by a centrifugal separator, or by the wet process, so that 
a large amount may be used with but little waste. 



THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 



167 



Some manufacturers prefer to mix mineral oil with lard 
or fish oil, and state the mixture is less expensive and works 
well. Prof. J. B. Denton has made experiments with mixtures 
and thinks that mineral or coal oil can be advantageously 
used. 

An excellent lubricant to use with a pump is by mixing 
together and boiling for one-half hour, 1^ pound sal soda, % 
pint lard oil, J^ pint soft soap, and water enough to make ten 
quarts. 

There is a difference of opinion as to whether the work 
should be moved against the cutter, or with it. But in most 
cases our experience and experiments show it is best for the 
work to move against the cutter, as shown at A, Figure 413^. 
When it moves in this way the teeth of the cutter, in com- 
mencing their 
work, as soon 
as the hard 
surface or 
scale is once 
broken, are 
immediately 
brought into 
contact with 
the softer 
metal, and 
when the 

scale is reached it is pried or broken off. Also when a 
piece moves in this way the cutter cannot dig into the work, 
as it is liable to do when the bed is moved in the direction 
indicated at B. When a piece is on the side of a cutter that is 
moving downwards, the piece should, as a rule, have a rigid 
support and be fed by raising the knee of the machine. 

Some work, however, is better milled by moving with the 
cutter. For example : To dress both sides of a thick piece (D) 
with a pair of large straddle mills, it might be well to move 
the piece toward the left, as the cutters then tend to keep 
it down in place instead of lifting it. 




168 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 

Again, in milling deep slots with a thin cutter or saw, it 
may be better to move the work with the cutter, as the cutter 
is then less likely to crown sidewise and make a crooked slot. 

When the work is moving with the cutter the table gib 
screws must be set up rather hard, for if the work moves too 
easily the cutter may catch, and the cutter orwork be injured. 
A counter weight to hold back the table is excellent in such 
milling. 

It is impossible to give definite rules for the speed and 
feed of cutters, and what is here said is only in the way of 
suggestions. The judgment of the foreman or man in charge 
of the machine should determine what is best in each instance. 

The average speed on wrought iron and annealed steel is 
perhaps forty feet a minute, which gives about sixty turns a 
minute for mills 2J£ inches in diameter. The feed of the work 
for this surface speed of the mill can be about 1% inches a 
minute, and the depth of cut say 1/16 of an inch. In cast iron 
a mill can have a surface speed of about fifty feet a minute, 
while the feed is 1% inches a minute, and the cut 3/16 of an 
inch deep ; in tough brass the speed may be eighty feet, 
the feed as before, and the chip 3/32 of an inch. 

As a small mill cuts faster than a large one, an end mill, 
for example, % inch in diameter, can be run about 400 revolu- 
tions with a feed of 4 inches a minute. 

For example of what may regularly be done under suita- 
ble conditions, we may mention that cutters 2^ inches in 
diameter used in cutting annealed cast iron in our works are 
run at more than 200 turns, or at a surface speed of more than 
125 feet, while the work is fed more than 8 inches a minute. 
The cuts are light, not more than 1/32 of an inch deep, and the 
work is short, from %. inch to 1 inch long. Two side mills 5 
inches in diameter running 50 turns a minute, dress both 
edges of cast iron bars % of an inch thick, with a feed of more 
than 4 inches a minute. 

An English authority, Mr. Geo. Addy, gives as safe speeds 
for cutters of 6 inches diameter and upwards : 

Steel, 36 feet per minute, with feed oi y?." per minute. 



THE MACHINIST AND TOOT, MAKER' S INSTRUCTOR. 169 

Wrought Iron, 48 feet per minute with feed of 1" per minute. 

Cast Iron, 60 feet per minute with feed of 1%" per minute. 

Brass, 120 feet per minute with feed of 2% 7/ per minute. 

And he gives as a simple rule for obtaining the speed: 
Number of revolutions which the cutter spindle should make 
when working on cast iron = 240 divided by the diameter of 
the cutter in inches. 

Mr. John H. Briggs, another English authority, states for 
cutting wrought iron with a milling cutter, taking a cut of one 
inch depth, which was a different thing from mere surface 
cutting, a circumferential speed of from 36 to 40 feet per min- 
ute was the highest that could be obtained with due considera- 
tion to economy, and to the time occupied in grinding and 
changing cutters; the feed would be at the rate of % inches 
per minute. 

Upon soft, mild steel, about 30 feet per minute was the 
highest speed, with 34 inch depth of cut and % inch feed per 
minute. Upon tough gun metal, 80 feet per minute, with J£ 
inch depth of cut and %inch feed. 

For cutting cast iron gear wheels from blanks pre- 
viously turned, and using in this case comparatively small 
milling cutters of only 3J£ inches diameter, the speed was 26J^ 
feet per minute, with J£ inch depth of cut and % inch feed per 
minute. 

Slotting cutters may often be run at a higher speed than 
other cutters of the same diameter, but with a wider face. 
Angular cutters must in some instances be used with a fine 
feed to prevent breaking the points of the teeth. 

Castings that are to be milled should be free from sand. 
They should be well pickled, and in some cases it is an advan- 
tage to have them rattled after being pickled. Where they are 
small and are to be finished rapidly it is also well to have them 
annealed. 

Forgings should be free from scale. They can be pickled 
in ten minutes in one part sulphuric acid and twenty-five parts 
boiling water, and if then rinsed in boiling water, they will 
dry before becoming rusty. 



170 



INDEX TABLE. 



NUM- 


NUMBER OF 




NUM- 


NUMBER OF 






BER OF 


HOLES IN 


NUMBER OF 


BER OF 


HOLES IN 


NUMBER OF 


DIVI- 


THE INDEX 


TURNS OF THE 


DIVI- 


THE INDEX 


TURNS 


OF THE 


SIONS. 


CIRCLE. 


CRANK. 


SIONS. 


CIRCLE. 


CRAINK. 


2 


ANY 


20 


65 


39 


24-39 




3 


39 


13 13-39 


66 


33 




20-33 


4 


ANY 


10 


68 


17 


10-17 




5 


ANY 


8 


70 


49 




28-4£ 


6 


39 


6 26-39 


72 


27 


15-27 




7 


49 


5 35-49 


74 


37 




20-37 


8 


ANY 


5 


75 


15 


8-15 




9 


27 


4 12-27 


76 


19 




10-lfr 


10 


ANY 


4 


78 


39 


20-39 




H 


33 


3 21-33 


80 


20 




10-20 


12 


39 


3 13-39 


82 


41 


20-41 




13 


39 


3 3-39 


84 


21 




10-21 


14 


49 


2 42-49 


85 


17 


8-17 




15 


39 


2 26-39 


86 


43 




20-43 


16 


20 


2 10-20 


88 


33 


15-33 




17 


17 


2 6-17 


90 


27 




12-27 


18 


27 


2 6-27 


92 


23 


10-23 




19 


19 


2 2-19 


94 


47 




20-47 


20 


ANY 


2 


95 


19 


8-19 




21 


21 


1 19-21 


98 


49 




20-49 


22 


33 


1 27-33 


100 


20 


8-20 




23 


23 


1 17-23 


104 


39 




15-39 


24 


39 


1 26-39 


105 


21 


8-21 




25 


20 


1 12-20 


108 


27 




10-27 


26 


39 


1 21-39 


110 


33 


12-33 




27 


27 


1 13-27 


115 


23 




8-23 


28 


49 


1 21-49 


116 


29 


10-29 




29 


29 


1 11-29 


120 


39 




13-39 


30 


39 


1 13-39 


124 


31 


10-31 




31 


31 


1 9-31 


128 


16 




5-16 


32 


20 


1 5-20 


130 


39 


12-39 




33 


83 


1 7-33 


132 


33 




10-33 


34 


17 


1 3-17 


135 


27 


8-27 




35 


49 


1 7-49 


136 


17 




5-17 


36 


27 


1 3-27 


140 


49 


14-49 




37 


37 


1 3-37 


144 


18 




5-18 


38 


19 


1 1-19 


145 


29 


8-29 




39 


39 


1 1-39 


148 


37 




10-37 


40 


ANY 


1 


150 


15 


4-15 




41 


41 


40-41 


152 


19 




5-19 


42 


21 


20-21 


155 


31 


8-31 




43 


43 


40-43 


156 


39 




10-39 


44 


33 


30-33 


160 


20 


5-20 




45 


27 


24-27 


164 


41 




10-41 


46 


23 


20-23 


165 


33 


8-33 




47 


47 


40-47 


168 


21 




5-21 


48 


38 


15-18 


]70 


17 


4-17 




49 


49 


40-49 


172 


43 




10-43 


50 


20 


16-20 


180 


27 


6-27 




52 


39 


30-39 


184 


23 




5-23 


54 


27 


20-27 


185 


37 


8-37 




55 


33 


24-33 


188 


47 




10-47 


56 


49 


35-49 


190 


19 


4-19 




58 


29 


20-29 


195 


39 




8-39 


60 


39 


26-39 


196 


49 


10-49 




62 


31 


20-31 


200 


20 




4-20 


64 


16 


10-16 


205 


41 


8-41 





THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 171 

INDEXING. 

The first office of the indexing head stock or spiral head 
is to divide the periphery of a piece of work into a number 
of equal parts, and in connection with the foot stock; it also 
enables the milling machine to be used for work sometimes 
done on planer centers and on gear cutting machines. 

As the index spindle may be revolved by the crank, and 
as forty turns of the crank make one revolution of the spin- 
dle, to find how many turns of the crank are necessary for a 
certain division of the work, or, what is the same thing, for 
a certain division of a revolution of the spindle, forty is di- 
vided by the number of the divisions which are desired. The 
rule then may be said to be, divide forty by the number of 
the divisions to be made and the quotient will be the number 
of turns, or the part of a turn, of the crank, which will give 
each desired division. Applying this rule — to make forty 
divisions, the crank would be turned completely around once 
to obtain each division; or, to obtain twenty divisions, it would 
be turned twice. When, to obtain the necessary divisions, 
the crank has to be turned only part of the way around, an 
index plate is used. For example: — If the work is to be 
divided into eighty divisions, the crank must be turned one- 
half way around, and an index plate with an even number of 
holes in one of the circles would be selected, it being necessary 
only to have two holes opposite to each other in the plate. 
If the work is to be divided into three divisions an index 
plate should be selected which has a circle with a number of 
holes that can be divided by three, as fifteen, eighteen, twenty- 
seven, etc., the numbers on the index plates indicating the 
number of holes in the various circles. 

The sector is of service in obviating the necessity of 
counting the holes at each partial rotation or turn of the 
crank, and to illustrate its use it may be supposed that it is 
desired to divide the work into 144 divisions. Dividing 40 
by 144 the result, 5-18, shows that the index crank must be 
moved 5-18 of a turn to obtain each of the 144 divisions. 
An index plate with a circle containing eighteen holes or a. 



172 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

multiple of eighteen, is selected and the sector is set to meas- 
ure off five spaces, or the corresponding multiple, ten spaces 
for example, in a circle with thirty-six holes. In setting 
the sector it should be remembered that there must always 
be one more hole between the arms than there are spaces 
to be counted or measured off. The required number of 
turns of the crank for a large number of divisions may be 
readily ascertained from the accompanying index tables, 
which correspond to the Brown & Sharpe Manufacturing 
Company's Milling Machines. 

If the angle of elevation of the spiral head spindle is 
changed during the progress of the work, the work must be 
rotated slightly to bring it back to the proper position, as 
when the spindle is elevated or depressed the worm wheel 
is rotated about the worm, and the effect is the same as if 
the worm were turned in the opposite direction, 

CUTTING SPIRALS WITH UNIVERSAL MILLING 
MACHINES. 

The indexing head stock or spiral head, as indicated in 
connection with the descriptions of the Universal Milling 
Machines, is used for cutting spirals, the flutes of twist drills, 
for example, as well as for indexing or dividing. 

A positive rotary movement is given to the work -while 
the spiral bed is being moved lengthways by the feed screw, 
and the velocity ratios of these movements are regulated by 
four change gears, shown in position in Figure 42 and known 
as the gear on worm or worm gear, first gear on stud, second 
gear on stud and gear on screw or screw gear. 

The screw gear and first gear on stud are the drivers and 
the others the driven gears. Usually those gears are of such 
ratio that the work is advanced more than an inch while 
making one turn, and thus the spirals, cut on milling ma- 
chines, are designated in terms of inches to one turn, rather 
than turns, or threads per inch; for instance, a spiral is said 
to be of 8 inches lead, not that its pitch is 1-8 turn per inch. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 173 




The feed screw of the spiral bed has four threads to the 
inch, and forty turns of the worm make one turn of the 
spiral head spindle; accordingly, if change gears of equal 
diameters are used, the work will make a complete turn while 
it is moved lengthways 10 inches; that is, the spiral will 
have a lead of 10 inches. But this lead is practically the 
lead of the machine, as it is the resultant of the action of the 
parts of the machine that are always employed in this work, 
and it is so regarded in making the calculations used in cut- 
ting spirals. 

In principle, these calculations are the same as for the 
change gears of a Screw Cutting Lathe. The compound 
ratio of the driven to the driving gears equals in all cases 
the ratio of the lead of the required spiral to the lead of the 



174 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

machine. This can be readily understood by changing 
the diameters of the gears. 

Gears of the same diameter produce, as explained above, 
a spiral with a lead of 10 inches, which is the same lead as the 
lead of the machine. Three gears of equal diameter and a 
driven gear double this diameter produce a spiral with a lead 
of 20 inches, or twice the lead of the machine; and with 
both driven gears twice the diameters of the drivers, the 
ratio being compound, a spiral is produced with a lead of 
40 inches, or four times the machine's lead. Conversely, 
driving gears twice the diameter of the driven produce a 
spiral with a lead equal to J4 the lead of the machine or 2y 2 
inches. 

EXPRESSING THE RATIOS AS FRACTIONS. 

Driven Gears Lead of Required Spiral. 

= ; or, as the product of 

Driving Gears Lead of Machine 

each class of gears determines the ratio, the head being 
double-geared, and as the lead of the machine is ten inches 
Product of Driven Gears Lead of Required Spiral 



Product of Driving Gears 10 

that is, the compound ratio of the driven to the driving 
gears may always be represented by a fraction whose nu- 
merator is the lead to be cut and whose denominator is 10; 
or, in other words, the ratio is as the required lead is to 10; 
that is, if the required lead is 20 the ratio is 20:10: or, to ex- 
press this in units instead of tens, the ratio is always the 
same as one-tenth of the required lead is to one. Fre- 
quently this is a very convenient way to think of the ratio; 
for example, if the ratio of the lead is 40, the gears are 4:1. 
If the lead is 25, the gears are 2.5:1, etc. 

To illustrate the usual calculations, assume that a spiral 
of 12-inch lead is to be cut. The compound ratio of the 
driven to the driving gears equals the desired lead divided 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 175 

by 10, or it may be represented by the fraction 12-10. Re- 
solving this into two factors to represent the two pairs of 

12 3 4 
•change gears, — = — X — • Both terms of the first factor are 

10 2 5 
multiplied by such a number (24 in this instance) that the 
resulting numerator and denominator will correspond with 
the number of teeth of two of the change gears furnished 
with the machine (such multiplications not affecting the value 

3 24 72 
of a fraction), — X — = — • The second factor is similarly 
2 24 48 

4 8 32 

treated, — X — = — > and the gears with 72 and 32 and 48 and 

5 8 40 

12 r 72X32 ) 
40 teeth are selected, — = < > The first two are the driv- 

10 (48X40) 
en and the last two the drivers, the numerators of the frac- 
tions having represented the driven gears, and the 72 is 
placed as the worm gear, the 40 as the first on stud, 32 the 
second on stud and 48 as the screw gear. The two driving 
gears might be transposed and the two driven gears might 
also be transposed without changing the spiral. That is, 
the 72 could be used as second on stud and the 32 as the 
worm gear, if such an arrangement was more convenient. 

From what has been said, the rules are plain. Note the 
ratio of the required lead to 10. This ratio is the compound 
ratio of the driven to the driving gears. Example: — If the 
lead of a required spiral is 12 inches, 12 to 10 will be the 
ratio of the gears, or divide the required lead by 10 and note 
the ratio between the quotient and 1. This ratio is usually 
the most simple form of the compound ratio of the driven 
to the driving gears. Example: — If the required lead is 40 
inches the quotient, 40-^-10 is 4, and the ratio 4 to 1. Having 
obtained the ratio between the required lead and 10 by one 
of the preceding rules, express the ratio in the form of a 



176 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

fraction, resolve this fraction into two factors; raise these 
factors to higher terms that correspond with the teeth of 
gears that can be conveniently used. The numerators will 
represent the driven and the denominators the driving gears 
that produce the required spiral. For example: — What gears 
shall be used to cut a lead of 27 inches? 

f 3 16) (9X8) 48X72 

27-10=3-2X9-5==^ -X— [ X \ \= 

( 2 16 J Ux8j 32X40 

From the fact that the product of the driven gears divided 
by the product of the drivers equals the lead divided by 10, 
or one-tenth of the lead, it is evident that ten times the 
product of the driven gears divided by the product of the 
drivers will equal the lead of the spiral. Hence, the rule: — 
Divide ten times the product of the driven gears by the prod- 
uct of the drivers, and the quotient is the lead of the resulting 
spiral in inches to one turn. For example: — What spiral will 
be cut by gears with 48, 72, 32 and 40 teeth, the first two 
being used as driven gears? Spiral to be cut equals 
10X48X72 

=27 inches to one turn. 

32X40 

Cuts that have one face radial are best made with an 
angular cutter, for cutters of this form readily clear the 
radial face of the cut, and so keep sharp longer and produce 
a smoother surface than when the radial face is cut in a verti- 
cal plane with a cutter where the teeth can have no side clear- 
ance from the work. 

The setting for these cuts must also be made before 
swinging the spiral bed to the angle of the spiral. 

THE UNIVERSAL MILLING MACHINE— Continued. 

(by the author ) 

From the preceding method of obtaining spirals as prac- 
ticed by the Brown & Sharpe Manufacturing Company, we 
find that the same rule will also apply to the milling ma- 
chine as constructed by others. For example: — Figure 43 is 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 177 




178 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

another style of the Universal Milling Machine, a portion of 
the index head only being shown. The feed screw A has two 
threads per inch, but there is a pair of bevel gears as shown 
at F G that drive the change gears, and F is twice as large 
as G; these bevel wheels are not to be changed, and B is 
called the screw gear. Now it can be seen that this gear 
near B will make four revolutions while the screw moves the 
table one inch, and as the index crank C makes forty turns to 
one turn of the spindle D, if all four of the change gears 
were of the same size, the table of the machine would move 
through a space of ten inches, while the index crank made 40 
turns or the spindle of the Universal head made but one turn, 
and consequently, the lead of the machine (figure 43) is also 
ten inches. 

Now, suppose we want to cut a spiral mill that is to have 
a lead of one turn in 60 inches; dividing 60 by 10 we have 
the ratio of the change gears, 6 to 1. 

Now, if we select for one pair of these gears 12 and 36 
or 24 and 72, or any number that is 3 to 1, dividing the ratio 
6 by 3=2; then any pair of gears that is 2 to 1 will answer for 
the second pair. 

In compound gearing always remember when you find 
the ratio of the gears wanted, and select one pair, that in 
order to find the second pair you must divide the ratio of the 
change gears by the ratio of the first pair selected; that is, 
we found the ratio of the gears were 6 to 1, and, taking 24 
and 72 for the first pair, which is 3 to 1, we divide 6 by 3 
and we have 2, which means that any number 2 to 1 will be 
right for the next pair, and therefore 64 and 32 will answer. 
Now, we can place the 32 on the screw and 64 on the stud to 
mesh with this gear, and the 72 on the worm with the 24 in 
gear with it; or, we can put the 24 on the screw and the 72 
in mesh with it and the 64 on the worm with 32 to engage 
with it; either way will be right. 

Let us take another example:— Suppose we want to mill 
a spiral of one turn in 28 inches, what gears should we use? 
We found the lead of the machine to be 10 inches, therefore 
dividing 28 by 10 we have 2 8-10, or 2 4-5, for the ratio of the 
gears. We will now try a pair that is 2 to 1, and see how 
we come out. Say 20 and 40, or 32 and 64 for the first pair 
of gears; now, 2 4-5=14-5, and any pair 2 to 1, or one gear 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 179 

being twice as large as its mate =2-1, which is equal to 10-5, 
and dividing 14-5 by 10-5 we have 1 4-10 for the ratio of the 
next pair, and taking 40 for one gear and multiplying this 
by 1 4-10, we have 56 for the next gear, so that 64 and 35 
will answer for one pair and 56 and 40 for the other; these 
gears are shown in position in the figure (43). Now let us 
prove this to be one turn in 28 inches: — As the screw is the 
driver, so is 40 the driver of 56, and as 32 is on the same 
stud with 56, then 32 is the driver of 64; then multiplying 
the product of the two driven gears by 10 and dividing by 
the product of the drivers, will be the lead of the spiral, thus: 
64=Driven 32=Driver 

56=Driven 40=Driver 

384 1280 

320 

3584 
10 



1280)35840(28", the answer, or the lead of spiral 
2560 



10240 
10240 



Example :— What spiral can we obtain from 72 and 48, 
as driven gears, and 28 and 40 as drivers? 

72=Driven 28=Driver 

48=Driven 40=Driver 

576 1120 

288 

3456 
10 

1120)34560(30.86", or nearly, the answer. 
3360 

9600 
8960 

6400 
6720 



180 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



^^K54*w£*Z!*z4£, 



The next thing is to find the 
angle to set the machine when cut- 
ting spirals, which is found in the 
following simple manner: Find the 
circumference of the work that is to 
be cut spiral, no matter what it is; 
take the outside diameter (except in 
'^p^case of a spiral gear, and then take 
the pitch diameter) and divide it by 
the length of the piece, which will 
give the tangent of the angle, and, in 
the table of tangents you will find 
the angle, or as follows: Figure 44 
represents a piece of work that is 28" 
long and 2%" diameter, that is to 
have a spiral of one turn in its length. 
The circumference of 2^"=9.032", 
and divided by 28=.3225; in the ta- 
ble of tangents, we find the angle 
17° 53' to correspond to this num- 
ber, which is the correct angle to' 
swivel the table. If the piece was- 
V/z" in diameter and the same length, we should then 
have a circumference of 7.854", nearly, and, di- 
vided by 28, we should have a tangent of .2805, which would 
then be 15° 40', as per table. Remember that this is a right 
angled triangle, or, in other words, the two lines marked 
circumference and length, as in figure 44, should always be 
at right angles to each other. 

A spiral mill 2>4" diameter is large enough for ordinary 
work and should have from 22 to 24 teeth, and should have 
a spiral of about one turn in 36". A cutter of this size and 
number of teeth will have about Yz" solid stock with a one- 
inch hole. 

For a spiral mill 3^" diameter I should give about 26 
teeth, and one turn in not less than 48 inches; a cutter of 
this size can be cut coarser than one of V/2" diameter, be- 




THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 181 

cause there will be more stock around the hole. If you want 
a good spiral mill for roughing purposes (that is, one that will 
cut fast), after the size has been finished in the lathe, take 
an ordinary cutting off tool, about one-eighth of an inch wide, 
with the corners rounded, and cut a thread of about two or 
three threads per inch in the opposite direction that you 
intend to mill the spiral. That is, if you wish to have a right 
hand mill, then you should cut this coarse thread in the 
lathe left hand. This is the best style of cutters we have for 
heavy or fast milling. 

If a spiral mill of about 2" diameter with a solid shank 
was required I should give it a spiral of about one turn in 
36", and should also make it right hand for the reason that, 
instead of pulling out of the spindle while finishing a cut, the 
tendency would be to force itself tighter in the spindle, and 
therefore less danger of marking the work. 

An angle of 10° to 12° will answer for most any ordinary 
size of spiral mills. 

In small end mills the teeth will have to be finer than 
large ones, to give them greater strength. 

I should give in end mills of y 2 " and */%' diameter eight 
teeth each; %", %" and 1" ten teeth each; \%* and l l / 2 " each 
twelve teeth, etc., and for a 1-34" diameter thin cutter, say 
3-16" to y thick, not less than sixteen teeth. 

If a pair of twin or straddle mills were wanted for gen- 
eral work, such as milling, bolt heads, etc., I would make 
them 4 inches diameter, y 2 " thick and not more than 32 
teeth, and for a pair of 5" diameter for general work, I 
should give them 36 teeth and make them Y% face; for a 
60° angular cutter, V$* diameter at the large end, I should 
give it 12 teeth, or, one of the same kind (60°) 2J4" largest 
diameter, 16 teeth; and one of 3" largest diameter (60°), 20 
teeth, etc. 

Always remember to have the teeth of cutters coarse 
enough so that in grinding the clearance the emery wheel 
will not touch the cutting edge of the next tooth. On most 
cutters the teeth on the periphery can be sharpened with 



182 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



wheels as large as 8" to 12" diameter, but on the sides of small 
cutters sometimes a wheel 2" diameter is too large. 

Milling cutters larger than 4 inches diameter and one 
inch face, and larger than 2]/ 2 inches diameter and 4-inch 
face, should have key ways: y%" wide and 3-32" deep is 
large enough for most purposes; the key ways should be 
slightly rounded in the corners to lessen the danger of crack- 
ing while hardening. 

I will now show you the shapes of cutters for a variety 
of purposes, and also the sizes, angles, etc., for ordinary 
work. I say ordinary, because we sometimes have to make a 
cutter twice the size or may be only one-half the size for 
some special class of work that we may have on hand. Fig- 
ure 45 represents a cutter for cutting the teeth of spiral mills, 





It is 12° on one side and 40° on the other side, as shown; the 
side marked 12° should always point to the center of the 
work, and, if two pieces having different diameters are to be 
milled, the table with the work will have to be moved in or out 
to suit that particular size, because setting it for any certain 
size it will not be right for any other diameter of work. The 
best way to set the work for this cutter is to raise the table 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 183 

until you think it high enough for the cutter when at the 
proper depth in the work, then, with a short scale or straight 
edge placed directly under the center of the cutter arbor 
and resting on the side marked 12°, with the center of tail 
stock also moved directly under the center of cutter arbor, 
the scale should point to the center of tail stock. The teeth 
of spiral mills on top should not be much over 1-32" thick 
when milled. 

Figure 46 represents an angular cutter of 60° for cutting 
the teeth in other cutters. This is generally used for 
cutting the teeth on the periphery of all ordinary cutters, 
except when the angles are very great. For instance, if the 
cutter shown at figure 48 was cut on both sides with a cutter 
of 60°, the teeth would be easily broken, and for this reason 
we will have to be governed accordingly. For the various 
shapes of cutters we use 60°, 65°, 70° and 75° cutters to cut 
them with, depending, as stated, on the acuteness of the 
angles. All cutters should have the teeth cut rather coarse 
when they are strong enough to stand it, for the reason that 
it will give more room for chips, and this is a very desirable 
matter with milling cutters. I would give the cutters (figures 
45-46) each 26 teeth for the diameter as shown. 

Figure 47 represents the neatest style of cutters for 
fluting taps; five sizes of cutters will answer for nearly all 
sizes. I would make them }i", y 2 " , %", J4" and 1" thick, 
and all of them 2^" diameter, but the corner of the small 
one should be rounded very little and the large one about as 
shown in the figure; the others should have the corners pro- 
portioned between these two. I would also give them each 
24 teeth. The width of land on all regular taps should be 
1-32 inch wide to every %" in diameter, as shown at B. This 
refers to standard sizes only. 

Figure 48 represents the shape of cutters for cutting the 
teeth in formed cutters, worm wheel-hobs, etc. 

Figure 49, represents the best style of milling cutters for 
milling the teeth in screw machine dies. I would give the 
former (figure 48) 32 teeth and the latter (figure 49) 24 teeth, 
for the diameters given. 



184 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 




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THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 



185 




Figure 50, which is also the 
same as figure 49, is shown in 
position for milling a hollow mill 
for screw machines. After set- 
ting the cutter in position, as 
shown, the table should be swiv- 
eled from about 20° for a y 2 " mill 
to about 30° for a iy 8 " mill. 
• For fluting reamers we some- 

i times use the cutter shown at fig- 

_ If ure 47, but only when the flutes 

gj^ are to be very coarse. For hand 
50 reamers they should be shaped as 
shown by the dotted line at A. 
All milling cutters should have 
the holes ground after hardening, 
and for this purpose I always allow five one-thousandths of 
an inch in the diameter; the sides always should be ground, 
as otherwise they will not run true. 

In these illustrations of cutters I have marked the thick- 
ness and also the diameters and angles as suitable for gen- 
eral purposes. 

For small shank milling I use a chuck as shown at figure 
51; it is composed of three pieces, A, B, C. The long taper 
of A is fitted to the machine spindle and bored large enough 
to suit a 34" shank mill nearly through. The back end is 
tapped %" and the spindle is bored large enough to admit a 
rod of that size extending from the back end to hold this 
chuck from any danger of slipping out and spoiling the work 
while in operation. These small mills are all made of Cres- 
cent tool steel (drill rods) and for a y 2 " mill you have only 
to cut off a piece of l / 2 " stock, say 5 inches long, as this 
is quite long enough for most purposes; cut the teeth 
on one end, say l 1 /*" , and, as the mills are parallel, you have 
the advantage of having them project just long enough for 
any particular class of work. 

In order to suit the different sizes, however, you will have 



186 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



to have one of these pieces, shown at B, for each size mill. 
This piece is reamed parallel, as shown by the dotted lines, 
and turned taper to suit the body of chuck nearly three-fourths 
of its length, and from there to the front end it is tapered in 



the opposite direct 



front, in order to draw the shell B tight on the shank of the 



mill; this shell B 



on and the nut C is also bored taper in 



s sawed in six equal spaces to within say 




THE MACHINIST AND TOOL, MAKEE's INSTRUCTOR. 187 

*4" from the end, each alternately starting from opposite 
ends, and in this manner the shell will close parallel on the 
work. The body of the chuck A and also the shell B should 
be made of tool steel. 

Figure 52 represents a special chuck that I made several 
years ago for milling bolt heads. The body of the chuck, 
which is about 2y 2 ,r through contains six 1*4" holes, a por- 
tion of which is cut off in the drawing for want of space. The 
lower portion of the chuck shown in dotted lines is screwed 
on the spindle of the Universal Head, and, being split, is 
held firmly in position by a screw. In order to get good re- 
sults with this chuck it is necessary that the holes should 
be accurately spaced; this is quite an easy matter, as they 
can be drilled, bored and reamed by indexing on the milling 
machine. These holes are 1J4" and we have split bushes of 
different sizes, so that all sizes of bolts can be milled up to 
and including V/\" . 

The bolts are held in position by the screw A and collar 
B, of which only one is shown. These collars and screws 
are all fitted in place before the holes C are bored, after which 
the bottom of B should have about one sixty-fourth of an 
inch taken off, so that they may clamp hard on the bush 
or bolt, as the case may be, so that they will not mark the 
work. 

In using this chuck the spindle of the Universal Head is 
placed in a vertical position with the face of the chuck up- 
right, as shown in the drawing, and six straddle mills about 
Zy 2 " or 4" diameter are used. The manner of using is as 
follows: The bolts, six in number, are placed in position 
with the heads projecting a little above the face of the chuck 
and the cutters set to just pass over without marking the 
chuck. The bolts 1, 2 and 3 are now milled on one side as 
shown by the dotted lines, and one-sixth of a turn is then made 
to the next bolt, and numbers 1 and 2 will then be milled on 
two sides, after which another one-sixth of a turn is made, 
and by this time No. 1 will be finished. At every one-sixth 
of a turn we can now take out a -finished bolt, and, of course,. 



188 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 

















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THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 189 

we also put in a blank every time we take out a finished bolt. 
Suppose that we want to mill some bolt heads 134" hex- 
agon (which can be represented by the holes) and the cutters 
are ground just V^' hub and face alike in width. The next 
thing to do is to find a spacing collar that will be' the right 
thickness to separate the two cutters shown at F. As there 
are six divisions represented by these holes, an angle, as 
shown, will be 60°, and G E being the radius of this angle, 
then D E is the sine of the angle of 60°, and the sine of 
60°=. 866 to a radius of 1"; then three times this would be 
the distance D E, or 2.598", and, taking from this the diameter 
of one of these bolts and the two one-half inch cutters, will 
leave .348" for the width of collar at F, as shown. 

STRAIGHT LINE INDEX DRILLING. 

It sometimes happens that we want a certain number of 
holes per inch drilled in a straight line; now, if it is possible 
to clamp this work on the milling machine bed or to hold 
it in the vise, we can very easily do a piece of work of this 
kind and know that it is right when done. 

Suppose that we want some holes spaced off to be exactly 
four per inch. As the screw has four threads per inch we 
know that every time the screw makes just one turn or y±" 
that it would be right for four holes, but in order to get this 
distance exact, we must gear up the machine similar to what 
we would if cutting a spiral, except that instead of turning 
the worm shaft crank for spacing we pull out the pin, being 
careful not to move it, but moving the index plate until the 
proper hole is reached and then securing it by the pin until 
the next move is to be made, etc. In other words, we move 
the table by compound gearing by the index plate, instead 
of the crank. 

If we have five holes per inch to drill, dividing 5 by 4 
threads on the screw, we have the ratio of the gears, or 1^4> 
which means that if we select a gear, say 40 teeth, and an- 
other one with V/\ times 40=50 teeth, then the next pair will 
be alike, and just one turn of the index plate will be correct. 



190 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 

If we want to drill ten holes per inch, dividing 10 by 4 
threads per inch, we have a ratio of 2y 2 for the gears. Now 
we will take 64 and 32 for one pair, which is 2 to 1, and divid- 
ing V/ 2 by 2 we have 1J4 for the ratio of the next pair, or 
say 48 and 60 teeth, etc. 

If seven holes per inch were required, dividing 7 by 4, 
we have 1J4 for the compound ratio; now we will take 32 and 
40 for one pair, which is a ratio of 1*4, or, in other words, 
one gear is one and one-quarter times as large as the other, 
or has a difference of one and one-quarter times as many 
teeth as the other; then for the next pair of gears we divide 
the compound ratio 1^4 by the ratio of the pair already se- 
lected and we have as follows: 1.75-^1.25=1.4 for the ratio of 
the next pair, then taking 40 for one of these gears and mul- 
tiplying it by 1.4 we have 56 for the last gear, and one turn 
of the index plate will move the table 1-7 of an inch. 

Suppose we prove this example. If we were to drill 100 
holes we would turn the index plate 100 times around in order 
to get that number of holes, and we have two pair of gears 
32 and 40 and 40 and 56. Now it is plain to see that the small 
gears are the drivers in both cases, and it makes no difference 
so long as we keep them in pairs as found, where we place 
them; then as we turn the index plate, we will put say the 40 
tooth gear back of it for the first driver, and its mate, 56, on 
the stud in gear with it; then the next small gear, 32, on the 
same stud in mesh with the 40 tooth gear on the screw; then 

100 holes=100 turns. 
40=first driver 

First driven=56)4000(71.4285 
392 

80 
56 

240 
224 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 191 



160 

112 

480 
448 



320 
280 



71.4285 

32=second driver. 



1428570 
2142855 



Second driven=40)2285.7120 



Threads on screw=4)57.1428==:revolutions of screw. 



14.2857=the number of inches the 
table has moved in drilling 100 holes. 

Holes per inch=r7)100=holes to be drilled. 

14.2857=the number of inches 
the table has moved. 

In calculating the gears for drilling 14 holes per inch, we 
divide 14 by 4, which is 3 J />, or the compound ratio of gears; 
then if we take any pair that is 2 to 1, say 64 and 32, and 
divide the compound ratio Syi by the ratio of the pair se- 
lected, which is 2 in this case, we have 1^4 as the ratio for 
the second pair, or say 48 and 84. 

It can readily be seen that if we should gear up for 
drilling seven holes to the inch with one turn, that we could 
also drill fourteen holes to the inch if we moved the index 
plate but one-half of a turn, or fifty-six holes per inch if 
moved only % of a turn, etc. Graduations can also be done 
in the same manner, but great care would be necessary if 
accuracy was required to take up all back lash in the running 
I parts, while the screw itself might not be in good condition 
to give the desired result. 



192 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



EXTRACTS FROM THE BROWN & SHARPE MFG. 

CO.'S TREATISE ON THE UNIVERSAL 

GRINDING MACHINE. 

(By permission). 

As the durability of a machine depends very largely upon 
the care of the operator, however well a machine may be 
constructed, if it is not properly cared for it will soon be- 
come unreliable. 

In all cases we recommend a good quality of oil in pref- 
erence to one of low grade. Sperm oil should be used on the 
internal grinding fixtures. 

The machine should be kept as clean as possible, and in 
no case should the bearings be allowed to "gum up." When 
bearings are opened and exposed for any purpose whatever, 
they should be carefully wiped off before they are closed 
again, to free them from any grit that may have found its 
way upon the surfaces. 

A loose fit between the wheel spindle and its boxes will 
produce imperfect work, and when very fine work is required 
the bearings should run nearly, if not quite, metal to metal. 
This necessarily will cause the boxes to heat, but in this case 
the heat is not injurious, for as the bearings are hard and the 
boxes bronze, the belt will slip, unless it is exceedingly tight, 
before abrasion can occur. 

All end motion should be taken out of the wheel spindle 
before the wheel is used on the work. 

A satisfactory emery wheel is an important factor in the 
production of good work. Too much, however, must not be 
expected of one wheel. A variety of shapes, sizes and grades 
of wheels are necessary to bring out all the possibilities of the 
grinding machine, the same as a variety of shapes and sizes 
of tools are necessary to obtain the best results from the 
lathe or milling machine. 

Our aim in grinding is usually to obtain an accurate or 
true surface, but as a true surface is almost always a ^ood 
surface it should be remembered that generally the same 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 193 

methods are employed, whether an exact size or a fine finish 
is the object desired. 

In selecting and using a wheel, we are governed by the 
character of the metal to be operated upon, the shape and 
size of the work and the degree of accuracy desired. We 
have to consider the size of the particles of emery in the wheel, 
the hardness of the wheel and its width. 

We also have to determine the speed at which the work 
is to travel or be revolved, and whether or not water is to be 
used. 

For the sake of clearness we refer separately to the 
various characteristics of wheels, but it should be borne in 
mind that a wheel should not be selected for a single charac- 
teristic, but that each of the essential elements is importantly 
affected by the others, and that all should be considered in 
choosing or using a wheel for any desired work. 

Wheels are numbered from coarse to fine; that is, a 
wheel made of No. 60 emery is coarser than one made of No. 
100. Within certain limits, and other things being equal, a 
coarse wheel is less liable to change the temperature of the 
work and less liable to glaze than a fine wheel. As a rule^ 
the harder the stock the coarser the wheel required to produce 
a given finish. For example, coarser wheels are required to 
produce a given surface upon hardened steel than upon soft 
steel, while finer wheels are required to produce this surface 
upon brass or copper than upon either hardened or soft steel- 
Wheels are graded from soft to hard and the grade is 
denoted by the letters of the alphabet, A denoting the softest 
grade. A wheel is soft or hard chiefly on account of the 
amount and character of the material combined in its manu- 
facture with emery or corundum, but other characteristics 
being equal, a wheel that is composed of fine emery is more 
compact and harder than one made of coarser emery. 

For instance, a wheel of No. 100 emery, grade B, will be 
harder than one of No. 60 emery, same grade. 

The softness of a wheel is generally its most important 
characteristic. 



194 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

A soft wheel is less apt to cause a change of temperature 
in the work or to become glazed than a harder one. It is 
best for grinding hardened steel, cast iron, brass, copper and 
rubber, while a harder or more compact wheel is better for 
grinding soft steel and wrought iron. As a rule, other things 
being equal, the harder the stock the softer the wheel required 
to produce a given finish. 

Generally speaking, a wheel should be softer as the sur- 
face in contact with the work is increased. For example, a 
wheel 1-16 inch face should be harder than one y 2 inch face. 
If a wheel is hard and heats or chatters it can often be made 
somewhat more effective by turning off a part of its cutting 
surface; but it should be clearly understood that while this 
will sometimes prevent a hard wheel from heating or chatter- 
ing the work, such a wheel will not prove as economical as 
one of the full width and proper grade, for it should be 
borne in mind that the grade should always bear the proper 
relation to the width. 

The width should be in proportion to the amount of ma- 
terial to be removed with each revolution, and as a wheel 
cuts in proportion to the number of particles in contact with 
the work, less stock will ordinarily be removed by a narrow 
wheel than by one that is of full width. The feed will also 
have to be finer if a narrow wheel is used. 

The quality of the work as a rule is improved by using 
a wheel of full width if the wheel is soft in proportion. 
Judgment should be exercised in deciding upon the width of 
wheel to be used, as sometimes the work is of such size and 
shape as to make it necessary to use a wheel with a narrow 
face. Where this is the case the wheel should, where strength 
will admit, be only that width throughout, and care should 
be taken that the grade is kept in the proper relation to the 
width. 

A wheel is most efficient in grinding just at the point 
before it ceases to crumble. The faster it is run up to this 
point the more stock will be removed and the more econom- 
ically the work will be produced. Occasionally, however, 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 1 95 

it is necessary to run a wheel rather slowly, as the more 
slowly it runs the coarser it cuts and the less likelv it is to 
change the temperature of the work. As a general rule, on 
any stock, the softer the wheel the faster it should be run.. 

Should a wheel heat or glaze it can often be made some- 
what more effective by being run more slowly. On the other 
hand, if it be too soft, it can often be made to somewhat 
better hold its size and grind straight by being run more 
rapidly. 

The surface speed of the work should be proportional 
to the speed of the wheel, that is, other things being equal, 
if the speed of the wheel is reduced the speed of the work 
should be reduced also. The desire is to nave the work 
revolve at such a speed as to allow time for the wheel to cut 
away the high points on the work. If the work is run so 
fast that there is no time given for the wheel to cut, but the 
work is simply crowded against the wheel, the tendency is 
for the wheel to follow the inequalities in the form of the 
work, and straight or round surfaces are not obtained. When 
the wheel is not free cutting and the pressure of the wheel 
against the work is sufficient to cause the work itself to 
spring cr to cause a slight movement of the oil upon the 
centers, the accuracy of the result is impaired. 

The iesire in accurate grinding is to have a free cutting 
wheel and to obtain the proper speeds so that the stock may 
be removed with the least possible amount of pressure, thus 
preventing a change of temperature in the work and allowing 
the high parts to be most speedily reduced. 

Thus far we have had in mind the selection and use of 
wheels for the comparatively small or medium sized work 
ordinarily ground on our machines. The requirements in 
grinding extremely large or long pieces are somewhat differ- 
ent. For example, in grinding a piece of steel thiee inches 
long, one inch diameter, on a Universal Grinding Machine 
we have indicated that the most absolutely accurate work 
would be accomplished by selecting a wheel only just hard 
enough to retain its size while passing six or eight times 



196 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

over the surface of the piece, and we have suggested that such 
a wheel should be run at a high rate of speed. We have con- 
sidered rapidity of production as more important than econ- 
omy of emery. If, however, we should attempt to use such 
a wheel to grind a piece of steel one inch diameter and three 
feet long, it is clear that before the wheel had passed over 
two of the three feet it would have ceased to cut. 

The problem now is to maintain the diameter of the 
wheel so as to take a uniform cut over a large area. Each 
particle of emery must be used as long as possible before be- 
ing thrown away. The particles may be used a longer time 
and are not so rapidly thrown away in a hard as in a soft 
wheel. Accordingly, one expedient in grinding large areas 
is to increase the grade of the wheel as the area increases, 
the speed of the wheel being reduced as the grade is increased. 

The loss of fine particles will not decrease the diameter 
of the wheel as rapidly as the loss of coarser or larger par- 
ticles. Thus another expedient is to use a finer wheel. 

A fine wheel can be relatively softer than a coarser wheel, 
and so with a fine one there need be less pressure between 
the wheel and the work, and there is more certainty of ob- 
taining an accurate surface. 

If a wheel is run rapidly the particles of emery soon be- 
come dull and have to be thrown away. To retard this loss 
it is w T ell to run the wheel more slowly as the length or area 
of the work increases. 

As the length or area of the work increases, the feed 
should be coarser, so that the wheel may travel the entire 
length or area of the piece while its diameter is practically 
unchanged. 

Water should be used on such classes of work as are in- 
juriously affected by a change in temperature caused by 
grinding. 

It should be used upon work revolved upon centers, as in 
this work a slight change in temperature will cause the wheel 
to cut on one side of the piece, after it has been ground ap- 
parently round. 



THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 197 

In very accurate grinding water is especially useful, for 
it should be remembered that the exactness of the work will 
be affected by a change in temperature which is not percep- 
tible to the touch. 

In very accurate grinding it is also well to use the water 
over and over again, as by so doing there is less difference 
between the temperature of the water and that of the work 
than if fresh water is used. For many purposes soda water 
is the most satisfactory, as it has less tendency to rust the 
work or the machine. 

For internal grinding it is especially important that a 
wheel should be free cutting and the work revolved so slowly 
as to enable the wheel to readily do its work. The wheels 
should generally be softer than for external grinding, as a 
much larger portion of the periphery is in contact with the 
work. Their small diameters make it impossible for the 
proper periphery speed to be obtained, and this must be 
considered in regulating the speed of the work. 

The wheels listed at the end of this chapter are those 
which our experience has shown to be suitable for the various 
purposes specified. Special cases may demand changes in the 
grade letter, but under ordinary circumstances the list should 
be accepted as a guide. 

The speed, diameter and width of wheels, and the num- 
ber of the emery cannot be changed without changing the 
grade and cutting qualities of the wheels. 

In mounting emery wheels there should always be elastic 
washers placed between the wheel and the flanges. Sheet 
rubber is best for this purpose, but soft leather will answer 
very well. 

In some cases manufacturers of emery wheels attach a 
thick, soft paper washer to each side of the wheel for this 
purpose, in which case no further attention is required in 
this direction. 

In all kinds of grinding the work should move in a direc- 
tion opposite to that of the wheel at the cutting point. 

To obtain good results when the grinding is done on the 



198 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 

centers it must be remembered that true center holes in the 
work, as well as true centers, are absolutely necessary. 

The face plates should be kept true. They can be readily 
ground in place on the machines. 

When slight tapers are desired for either external or in- 
ternal grindings, the swivel table is set to the proper angle. 

When more abrupt tapers are wanted for work ground 
on centers or for internal surfaces, the wheel slide is set to 
the proper angle. By placing the wheel slide and the swivel 
table at the proper angles, two tapers, for either external or 
internal work, may be obtained without changing the set- 
tings of the machine, the one automatically by the longitu- 
dinal movement of the table, the other by operating the cross 
feed by hand. 

When an abrupt taper, similar to that shown in Figure 1, 
is to be ground, the swivel table remains parallel to the ways 
of the bed as in plain grinding, but the wheel bed is set to the 
angle, which brings the line of motion of the wheel slide, 
when operated by the cross feed, parallel with the taper to be 
obtained The wheel platen is set at right angles with the 
line of »i ovement of the wheel slide, indicated by the arrow, 
and the face of the wheel is thus brought parallel with the line 
of the desired taper. The work is revolved by the dead center 
pully, as shown in cut, and the wheel is moved over the sur- 
face of the work by the cross-feed. 

The method of grinding two tapers with one setting of 
the machine when one of the tapers is not more than ten 
degrees is shown in Figure 2. 

For grinding the slight taper the swivel table is set over, 
and for grinding the more abrupt taper the wheel bed is set 
as in Figure 1, but the wheel platen is here set to bring the 
face of the wheel parallel with the longest surface to be 
ground. 

Were the abrupt taper longer than the slight taper it 
would be well to set the wheel platen as in Fgure 1, so that 
the face of the wheel would be parallel with the line of taper. 
In obtaining the angle at which the wheel bed is to be set 



* THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 199 




200 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 




Wg2 



when the swivel table has been set over, it should be remem- 
bered that the angle must equal the sum of the two tapers. 
The abrupt taper is ground by feeding the wheel across the 
work by hand. The slight taper is ground while the table is 
fed automatically. 

When, as suggested by the cut, a spindle and box are 
both to be ground, the box is ground first and the spindle 
is fitted to it. For convenience in fitting the work, the box 
may be placed as shown by the dotted lines and supported 
so that it will not touch the spindle as the latter revolves. The 



THE MACHINIST AND T00E MAKER'S INSTRUCTOR. 



201 




^^ 3 



spindle thus need not be removed from the centers when it 
is necessary to try it in the box. 

In grinding the box the machine is set as shown in Figure 
3. Provision is made for grinding the slight taper by setting 
the swivel table, and for grinding the abrupt taper by swivel- 
ing the wheel bed. 

Thin washers and cutters are conveniently mounted on a 



202 THE MACHINIST AND TOOD MAKEE'S INSTRUCTOR. 

special chuck furnished with the machine. This chuck holds 
the cutters by the hole in the center and should be used in all 
disk grinding where both sides of the pieces must be parallel. 
For most disk grinding it is also more convenient than a 
common chuck and more accurate results are generally ob- 
tained by its use. Thin saws are held this way and are 
ground concave or thinner at the center than at the teeth, to 
give the proper clearance. 

Angular or taper cutters may be ground by holding them 
on a mandrel, but the swivel table must be set over for the 
proper taper. When the taper of the cutter is greater than 
two inches per foot, the feeding may be done with the cross 
feed, and the wheel set as shown and explained in connec- 
tion with Figure 1. 

The side teeth of straddle mills may be ground on a 
Universal Grinding Machine, but the machine is not recom- 
mended for such a purpose, as, in order to obtain the proper 
clearance, the wheel stand must be elevated so that the center 
of the wheel will be above the tooth rest, the tooth rest in 
this case having to be set as high as the center of the head 
stock spindle. 

Where many cutters of this class have to be ground, or, 
in fact, for manufacturing establishments having a great deal 
of tool grinding, we would recommend the Brown & Sharpe's 
special tool and cutter grinding machines as more convenient 
and economical. 

The following list of emery wheels, as made by the Nor- 
ton Emery Wheel Company of Worcester, Massachusetts, 
to be used in connection with the Number 2 Universal Grind- 
ing Machine Improved (Brown & Sharpe), is recommended. 

To grind hardened steel spinales, etc., shape 3 (plain 12" 
diameter, J^" face), emery 60, grade K, mixture 14- A, speed 
1,300 to 2,000; also emery 60, grade M, mixture 13- A, speed 
1,300 to 2,000. 

To grind soft steel, shape 3, emery 100, grade L, mixture 
14-A, speed 2,000. 

To grind long or large soft steel pieces, shape as before, 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 203 

emery 100, grade N, mixture 5- A; also emery 100, grade O, 
mixture 6-A. 

To grind collars, disks, saws, etc., shape 1, 13 or 14 (plain 
6" to 7" diameter, and from 34" to y 2 " face), Georgia corun- 
dum 36, grade J, mixture X-15-C, speed 3,000. 

To grind cast iron spindles (solid), shape 3, (12" diameter 
y 2 " face), emery 60, grade M, mixture 13- A, speed 1,300 to 
2,000. 

To grind hollow cast iron rolls, shape 3, emery 60, grade 
F, mixture 15-A, speed 1,300 to 2,000. 



WHEELS FOR INTERNAL GRINDING, 

TO ROUGH OUT FOR ACCURATE WORK BUT CANNOT BE 

CROWDED. FOR HARD AND SOFT STEEI* 

AND CAST IRON. 

Shapes 82, 83 and 84 (from y 2 " to J4" diameter, Y A " face 
and yi" hole), emery 46, grade G, mixture X-12-A, speed 
13,400. 

Shapes 84, 85 and 86, (from J4" to 1* diameter, %" hole 
and 34" face), emery 46, grade G, mixture X-12-A, .speed 
12,200. 

Shapes 87, 88 and 89, (from 1" to iy 2 " diameter, y 8 " face 
and from y to y%" holes), emery 46, grade G, mixture X-12- 
A, speed 11,200. 

Shapes 90 and 91, (from 2" to 2^" in diameter, y s " face 
and J4" hole), emery 46, grade G, mixture X-12-A, speed 
8,050. 

To grind for finishing after roughing wheels have been 
used: 

Shapes 82, 83 and 84, emery 120, grade F, mixture X-10- 
A, speed 13,400. 

Shapes 84, 85 and 86, emery 120, grade F, mixture X-10- 
A, speed 12,200. 

Shapes 87, 88 and 89, emery 120, grade F, mixture X-10- 
A, speed 11,200. 

Shapes 90 and 91, emery 120, grade F, mixture X-10-A, 
speed 8,050. 

For internal grinding the Special Fixtures are used. 



.204 THE MACHINIST AND T00X, MAKER'S INSTRUCTOR. 



CHAPTER V. 



MECHANICS. 

On this subject I shall treat more particularly on such 
matters as concern the mechanic in his surroundings. 

QUESTIONS UPON THE PRINCIPLES OF THE LEVER. 

At one extremity of a straight lever, whose length is 6 
feet, a weight of 10 pounds is suspended; at the distance of 
5 feet from the point of suspension a fulcrum is placed. 
What weight must be suspended from the other extremity of 
the lever to keep it in equilibrium (balanced)? 

If 10 pounds is suspended 5 feet from the fulcrum, then 
at one foot from the fulcrum on the opposite end it will take 
5 times as much to balance it, or 50 pounds, the answer. 

A lever is 30 feet long; at what distance from the fulcrum 
must a weight of 200 pounds be placed so that it may be 
supported by a power able to sustain 50 pounds? 200-^50=4, 
then 30-^4=7^2 feet, the answer. 




In a lever, as shown in figure 1, a weight of 100 pounds 
is suspended at A. What will be the required weight at B to 
hold this weight in equilibrium? Both arms being of the 
same length from the fulcrum C, but while A is horizontal 
B is inclined at an angle of 45°, as shown. 

The cosine of 45 o =.707, then 1-f-. 707X100= 141. 3 pounds, 
the answer. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



205 



U. 30" ~^ — 50" 



€ 



In the lever, as shown at figure 2, A is the fulcrum, B 
the weight, and the power is applied at C in the direction of 
the arrow. Required, the power at C to support a weight of 
60 pounds in equilibrium at B, as shown. 

In this case the weight at B multiplied by the distance 
A B must equal the power at C multiplied by the distance 
A C, or thus: Weight at B=60 pounds, distance A B= 
30", then 60X30=1800; the distance A C is 30"-f50", 
or 80", then 1800-f-80"=22.5 pounds, the power at C to bal- 
ance a weight of 60 pounds, as shown at B. The pressure 
on the fulcrum at A is equal to the difference between the 
weight suspended at B and the power at C; then, as the 
weight at B=60 pounds and the power at C— 22.5 pounds, 
60—22.5=37.5 pounds as the pressure on the fulcrum. 



!«e- 30". 



50" 



In figure 3 the power is represented between the fulcrum 
and the weight. Required, the power at B to support a 
weight of 60 pounds suspended at C, as shown. 

In this, or similar examples, we multiply the weight by 



206 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



the distance A C and divide by the power multiplied by the 
distance A B; or thus: 60x80"=4800; now, as the power is 
not known we divide 4800 by the distance A B, or 30", thus: 
4800-^30"=160 pounds, or the power at B to just balance a 
weight of 60 pounds at C. In this example, as the power is 
greater than the weight, the pressure at the fulcrum is up- 
wards, and is equal to the difference between the power and 
the weight, thus: 160—60=100 pounds. 

The safety valve also comes under the head of levers, and 
the calculations are similar to figure 3. In figure 4 let A be 




the fulcrum of the lever, B the valve, 3" from the fulcrum, 
and D a weight of 40 pounds suspended 25" from the valve 
or 28" from the fulcrum; the valve is 3" diameter. Required, 
the steam pressure on the valve to just balance the weight. 
(The weight of valve and lever is not taken into considera- 
tion). The distance A C is 28", and the weight 40 pounds, 
then 28X40=1120; the valve is 3" in diameter, 3X3=9X-7854 
=7.07 square inches, or the area of the valve; this being 3" 
from the fulcrum, we multiply it thus: 7.07X3=21.21, and 
1120-^21.21=53.2 pounds pressure, the answer. 

Now, suppose that our safety valve is 4" diameter and 
Sy 2 ff from the fulcrum, and we want to blow off at 100 pounds 
pressure, the weight is 90 pounds; how far from the fulcrum 
will it have to be suspended? 

The valve is 4" diameter, then 4X4=16, and .7854X16= 
12.56, or the number of square inches of area in the valve, 
and this, multiplied by the distance from the fulcrum, 3^", 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



207 



thus: 12.56X3.5=43.9, and again by the pressure required, 
43.9X100=4390; then 4390-^90=48.8 inches, nearly. This 
weight, of course, is supposed to balance the pressure on the 
valve, and any excess in the pressures will lift it. The weight 
of valve and lever has not been taken into consideration. 

We have another safety valve 3^" diameter and placed 
254" from the fulcrum, and we wish to maintain a boiler pres- 
sure of 65 pounds; the extreme length of lever from the 
fulcrum is 35 inches. Required, the weight, if suspended at 
the end of the lever. 

The valve is 3.5" diameter, then 3.5"X3.5=12.25X.7854= 
9.62, the area in square inches of the valve; then, 9.62X2.75, 
the distance from the fulcrum, =26.45, and multiplied by the 
pressure, 65 pounds, =1719.25; this divided by the length 
of lever 35' =49.1 pounds, as the required weight on the lever. 

It must be remembered that the diameter of the valve 
in all cases must be taken from where the steam acts directly 
against it, and not from the largest part of the valve. 



h 3 




We have a safety valve (Fig. 5); the fulcrum A is 3 inches 
from the valve B, the weight C is 65 pounds, and the weight 
of the lever D is 10 pounds and its center of gravity (of the 
lever) is 9 inches from the fulcrum A; the valve is 3^ inches 
diameter at the point of pressure, and we wish it to lift when 
the pressure is 70 pounds per square inch; how far from the 
fulcrum will we have to place the weight? 

The valve being 3^ inches diameter, then 3.5X3. 5X-T854 
=9.62 inches, area of the valve, or the number of square 



208 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 



inches; then, 9.62x70, the steam pressure, and again by 3", 
the distance A to B, =2020.2. The iever weighs 10 pounds, 
and its center of gravity is 9 inches from A; then, 10x9=90, 
and 2020.2—90=1930.2, and this divided by the weight, thus: 
1930.2-^65=29.7 inches, nearly, the answer. 

Or, if the lever had a notch filed into it 29.7" from the 
fulcrum and our weight had been lost and we wished to know 
how large a weight would be required under the same condi- 
tions, we would divide 1930.2 by 29.7 inches, and this would 
give the answer, or the weight required. 






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We have a reel, attached to a lever at B, (Fig. 6), that 
swings on a stud shown at A, the long arm of which is 38" 
from the fulcrum. The short arm is in the form of a segment 
of a gear and is 16" from the pitch line to the fulcrum. To 
turn this lever, we have a crank attached to the pinion D, 16" 
long; the pinion is 2 1-6" diameter at the pitch line. What 
force will be required at E to balance a weight of 153 pounds 
suspended at B in a horizontal position, disregarding friction 
and weight of the moving parts? 

In this example the force required at C will be as much 
greater than at B as the length from A B is greater thar 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



209 



A C; therefore, dividing 38 by 16, which is 2j^, and, as the 
weight at B is 153 pounds, multiplying this by 2j£ we have 
363.4 pounds to be overcome at C. But as the crank E is 
16" long, and the pinion that it turns is but 1 1-12" radius 
(one-half the diameter), we gain a leverage equal to the dif- 
ference between the radius of the pinion and the radius of the 
crank; then, dividing 16" by 1 1-12", or 1.083", and the result, 
14.7, is the ratio of the power gained by the difference in the 
radius of the crank to that of the pinion; then, dividing 363.4 
by 14.7 and the result, 24.7 pounds, will be the force required 
on the crank E to balance 153 pounds suspended at B. (Dis- 
regarding friction, etc.) 

A force of 36 pounds would be a safe estimate in this 
case, including friction, to lift the weight. 

It must be remembered that if the weight is in any other 
than a horizontal position from the fulcrum on which it moves 
that the distance A B, will be equal to the cosine of the 
angle, as shown in the figure. Thus, suppose that B was in- 
clined at an angle of 45° (shown at F), then as the cosine of 
45° =.707, and multiplied by 38" will be 26.8", for the distance 
to be considered for A B instead of 38". 

THE INCLINED PLANE. 




We have a barrel weighing 300 pounds, (Fig. 7), and we 
want to roll it into a wagon that is 45 inches high by using 
a plank (A) which is ten feet long; what force will be necessary 
to roll it up the inclined plane as shown when the pull is par- 
allel with the line A? 

300X45=13500 and 13500-^120" (10 feet) =112.5 pounds. 



210 THE MACHINIST AND TOOD MAKEE's INSTRUCTOR. 

This is the force necessary to hold the barrel on the plank, 
and if there was no friction, any excess of pressure ever 112.5 
pounds would roll the barrel. Or, multiply the weight by 
the perpendicular of the angle formed and dividing by the 
hypothenuse will give the answer. 

From the preceding example, knowing how much force 
you could exert without fatigue, and having a barrel of oil, 
or anything similar, to roll into a wagon, if the weight was 
known, you would know how long a plank to select for the 
purpose. Suppose you were alone and wished to roll a barrel 
weighing 250 pounds into a wagon that was 45 inches high, 
and 100 pounds was about as much as you cared to exert; 
what would be the length of the board required? 

250-^-100=2^, then 45x2^=112.5 inches, or nearly 10 
feet. This would not be quite enough to overcome the fric- 
tion, and we would get one a little longer for the purpose. 

On the same principle, if you were to draw a wagon 
weighing 200 pounds up a hill that was 60 feet high in 300 
feet in length, you would not only have the friction to over- 
come, the same as on a level road, but you would also add 
to this 60-300 of 200 pounds, or 40 pounds. 

THE SCREW. 

We have a screw, the pitch is 1 inch (or the distance be- 
tween two threads) and a lever 30 inches long, measuring 
from the center of the screw (or its axis); what pressure can 
be produced by this screw, if we apply a force of 45 pounds 
at the end of the lever? 

In this or similar examples we find the circumference of 
the circle described by the end of the lever, or where the 
power is applied, which in this case would be, thus: 30 inches 
radius =60 inches diameter, then 60X3.1416=188.5, and 188.5 
X45, the force at the end of the lever, =8482.5, and this divided 
by the pitch will give the answer. As the pitch in this case 
is one inch, then 8482.5 pounds is the answer. Now, if this 
screw was two inches pitch then we would have a pressure 
of one-half as much, or 4241.25 pounds. But if the screw 



THE MACHINIST AND TOCTL MAKER'S INSTRUCTOR. 211 

was % inch pitch, or four threads per inch, then we would 
have a pressure of four times as much, or 33930 pounds. 



WHEEL AND AXLE. 




In this example, Fig. 8, 
we have two pulleys or 
drums on the same shaft. 
A is 24" diameter and B is 
7" diameter; a weight of 40 
pounds is suspended from 
the largest pulley, shown 
at C; what weight will be 
required at D to balance the 
weight at C? A being 24" 
and B 7", then 24-^7=3 3-7, 
and 40 times 3 3-7=about 
137 pounds, the answer. 



WHEEL GEARING. 




In this example, Fig. 9, (compound gearing), A is 
the driving gear with 20 teeth, B 62 teeth and C 12 teeth 



212 THE MACHINIST AND TOOT, MAKER* S INSTRUCTOR. 

(B and C are both on the same shaft or stud), and D has 100 
teeth; now if we turn the gear A with a force equal to 15 
pounds on the pitch line, what will be the pressure on the 
pitch line of D? 

If A exerts a pull of 15 pounds on the circumference of 
B, then the force acting on the circumference of D will be as 
62 is to 12 times 15 pounds, thus: 62-4-12=5 1-6, which mul- 
tiplied by 15=77.5 pounds, nearly, the answer. 

It can easily be seen that if the gear C was of the same 
diameter as B, that the force exerted on the periphery of D 
would not change. In other words, it would be 15 pounds, 
and if it was one-half as large we would double the leverage, 
and consequently would have a pressure of 30 pounds on D. 
But if C was twice as large as B, then we would have but 
one-half of 15 pounds, or iy 2 pounds. 

Suppose we have a machine on which there is a com- 
bination of gears, and also a drum operated by a crank, as 
follows: On the first shaft we have a pinion 4" diameter and 
a crank 16" long (or radius). On the second shaft there is a 
gear 15" diameter and also a pinion 4" diameter, and on 
the third shaft there is a gear 15" diameter, and a drum 5 
inches diameter, with a rope wound around it. These 4" pin- 
ions in both cases mesh with the 15" gears. If we exert a 
force of 50 pounds on the crank, what will be the weight that 
we could hold in suspension attached to the rope? 

In this example we will first find the ratio between each 
pair of gears, thus: 15-^4=3 J4, and as there are two pairs 
alike, then we multiply these two ratios together, thus: 3.75 
X3.75=about 14 for the compound ratio of the gears. We 
next find the difference between the radius of the crank and 
the radius of the drum, thus: 16-^-2^=6.4, and multiplied 
by the compound ratio 14=89.6, and this again multiplied by 
50, the force on the crank =4480 pounds, the answer. 

Or, as follows, which is the usual way. 15"Xl5"=225", 
and 4"X4"=16"; then 225"-^-16"=14 ratio, which multiplied 
by 16", the leneth of crank=224, and agrain by 50, the force 
on the crank=11200, divided by one half the diameter (radius) 
of the drum=4480 pounds. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 213 




THE SCREW AND GEAR. 

We have a machine combining the following parts, as 
shown in Fig. 10, in which A is a crank, 16" radius, o- length; 
B is a worm of 4 threads per inch; C a worm wheel 12 inches 
diameter on the pitch line. On the same shaft with the worm 
wheel is a pinion D, 3 inches diameter (pitch line) in mesh 
with the gear E, 16 inches in diameter. On the same shaft 
with this gear (E) is a small pulley F, 5 inches diameter, with 
a rope wound around it, as shown at G, making no allow- 
ance for either the friction or diameter of the rope; what 
weight can we support at G by a force of 30 pounds applied 
on the crank at A? 

The crank A being 16" long will describe a circle of 32" 
in diameter, the circumference of which is 100.5 inches. Now, 
as the worm is Y\" pitch (4 threads) then the crank A will 
pass through four times as great a distance, or 402 inches, 
while the worm, or what is the same thing, the worm wheel, 
moves but one inch (on the pitch line). 

As the worm wheel C is 12" diameter and the pinion D 
but 3" diameter, we gain a leverage of 3 to 1; then 462X3= 



214 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



1206. It is plain that whatever may be the power gained at 
D will be imparted to E, since D drives E. But as E is 16 
inches in diameter and F is but 5 inches, we gain accordingly; 
thus: 16—5 =3 1-5; then 1206X3 1-5=3859.2, which multiplied 
by the force on the crank of 30 pounds=U5776 pounds, the 
answer. 

THE HYDRAULIC PRESS. 

Suppose we have a small hydraulic press, the lever of 
which is 24 inches in length to the fulcrum; this lever is 
worked by hand and is attached to the plunger, which is but 
three inches from the fulcrum and also between it and the 
power. Suppose this plunger to be 24 of an inch in diameter 
and the ram to be 12 inches diameter; by exerting a force of 
40 pounds on the lever, what pressure can be obtained by 
the ram? 

As the power applied is 24 inches from the fulcrum and 
the plunger is but 3 inches, we gain a ratio of 8 to 1 by 
moving the lever 8 inches to obtain 1 inch of plunger. The 
diameter of the plunger is $4", then .75X-75X3. 1416=442 
thousandths of an inch (area of the plunger), and the ram 
12"X12"X3.141«3=113" (area of the ram); then 113-=-.442== 
255.6; or, in other words, the area of a cross section of the 
ram is more than 255 times as great as that of the plunger; 
then multiply 255.6 by the ratio of 8 and again by the force 
of 40 pounds on the lever, and we have as follows: 40X8X 
255.6=81792 pounds, the answer. 

TO FIND THE BRAKE HORSE-POWER OF AN 
ENGINE. 

To find the bra*ve horse-power of an engine proceed as 
follows: Clamp the friction brake on a pulley placed on the 
crank shaft of the engine with the arms A A in such a posi- 
tion that the ends B of the arms which meet will be in a 
horizontal position with the center of pulleys shown in figure 
11; the end of arms at B are to rest on a scale blocked up in 
order to be in this position. Now we find that the pressure 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 215 



on the scale is iust 70v pounds, and the distance from B to 
the center of pulley or brake is 6 feet; then multiplying to- 
gether, thus: 700 y 6=4200, and dividing this by the radius 
of the pulley on which the brake is secured, we have as 
follows: 4200-1-2=2100; this is the resistance upon the cir- 
cumference of the pulley. Now, if this pulley revolves 125 
times per minute, we must find how many feet per minute 
the circumference of the pulley is making. Then 3.1416X4= 
12.56, the number of feet around the pulley, and this multi- 
plied by the speed, thus: 12.56X1-5=1570; this equals the 
number of feet per minute the pulley travels at the circum- 
ference, and this again multiplied by the resistance on the 
pulley as follows: 1570X2100=3297000, or the foot pounds 
per minute; then dividing by 33000 we have as follows: 
3297000-^33000=99.9 horse-power of the engine (brake). 

It must be remembered that the brake should be tight- 
ened sufficiently on the pulley until the speed of the engine 
is about to slack; this can easily be seen by the aid of a 
speed indicator. 

HORSE-POWER. 

The horse-power of an engine is found in the following 
manner. Example: We have an engine 24 inches diameter 
of cylinder, 40 inches stroke of piston, making 120 revolutions 
per minute, with steam at 75 pounds pressure per square inch; 
required, the horse-power. We first find the area of the piston 



216 



THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 



by squaring the diameter and multiplying it by .7854, thus: 
24X24X.7854=452, and multiplying this by the pressure of 
steam=:33900; multiplying this again by the stroke in 
feet per minute, which is as follows: as the stroke is 40 inches, 
at every revolution of the engine the piston will travel 80 
inches, then multiplying this by 120 (revolutions per minute) 
and we have 9600 inches, or 800 feet per minute; then 33900 
X800=27120000 foot pounds per minute. Then as a horse- 
power is equal to the work of 33000 pounds done in one min- 
ute, we divide 27120000 by 33000 and we have 822 horse-power 
for the answer. This is called the nominal horse-power; the 
indicated horse-power is found in the same manner, excepting 
that the steam pressure (the mean effective) is found by 
means of the indicator, which is always less than the boiler 
pressure. 

PRESSURE ON THE GUIDES OF AN ENGINE. 

Suppose we have an engine the cylinder oi which is 30 
inches bore, and 40 inches stroke, and we want to know the 
greatest pressure on the guides during a revolution, using 
100 pounds pressure of steam per square inch and calling the 
connecting rod 10 feet long between centers. 




In figure 12, let A B, 10 feet or 120 inches be the con- 
necting rod; B C=the length of crank, or one-half of the 
stroke. In the figure B C is drawn perpendicular to the 
line A C 5 because the greatest pressure is on the guides 
when the crank is at half stroke, as shown. Now, the cylin- 
der is 30 inches bore, and consequently the area is 30X30X 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 217 

7854=706 inches; multiplying this by the pressure of 100 
pounds=70600 pounds pressure on the piston, and conse- 
quently on the line A C; 70600 multiplied by the distance 
(20") B C, and divided by the length A C, will give the 
pressure on the guides. 

In the figure (12) we have a right angled triangle, as 
shown, the radius of which is 120 inches; B C is the 
sine of the angle B A C; then 20 inches (the sine) divided 
by the radius of 120 inches=.166, or the sine to a radius of 
1 inch, and, in the table of sines, we find this to correspond 
with the angle of 9° 33'. The cosine of this angle=.986; 
then multiplying this by the radius of 120", thus: .986X120'' 
=118.3 inches, or the length oi A C. Then 70600X20= 
1412000 and divided by 118.3=11935+ pounds, the answer. 

TO FIND THE DIAMETER OF A SHAFT FOR 

TRANSMITTING POWER, WHEN THE 

HORSE-POWER OF THE ENGINE 

IS KNOWN. 

Example: Suppose in a factory we have an engine 150 
horse-power which is to run at a speed of 175 revolutions per 
minute. Required, the diameter of the main line of shafting. 

175)150.00000(.857142 
140 

10 00 



1250 
1225 

250 
175 

750 

700 

500 



218 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 





.857142(.94+ 
729 




128142 
9 2 X300=24300 
9X30X4 = 1080 

4 2 = 16 101584 


.95+ 
2.9 


25396 26558 


855 
190 





2.755 inches, the answer. 

We first divide the horse-power of the engine by the 
revolutions it is to run per minute and extract the cube root, 
which in this case is .95, then multiply this by the constant 
2.9, which gives about 2J4 inches for the diameter of the 
shaft. This is for good ordinary steel, but if it is to be 
of wrought iron, it should be one-eighth (J^) larger, thus: 
y 8 of 2.755"— 344", and, added together =3.099", the answer, 
if made of wrought iron. 



IMPACT OR COLLISION OF BODIES. 

Let us now for a moment examine into the force of a 
shot fired from one of our modern 12-mch guns. Suppose a 
solid shot, weighing 1,100 pounds, is fired at an object and 
strikes it with a velocity of 1,500 feet per second; then the 
mass of this body, which is the weight divided by 32.2, mul- 
tiplied by the velocity in feet per second will be the answer, 
provided it takes just one second from the time the shot 
strikes the object until the momentum of the ball is stopped, 
or, as we say, destroyed; but we know that it does not take 
a second to stop it, and we will suppose that it takes but one- 
fiftieth (1-50) part of a second to stop it, then the force will 



THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 2lS k 

be as follows: 1100-^-32.2=34.16, which multiplied by the 
velocity equals 41240, which would be the answer, as explained 
if it required one second of time from the moment of striking^ 
until the momentum was destroyed; but if it only takes one- 
fiftieth of a second, then the force (average) will be 50 times 
as much, or 2062000 pounds, the answer. Or, if it takes the 
1-100 part of a second, then the answer will be 41240X100= 
4124000 pounds. 

The force of a railway train or any other object can also 
be found in a similar manner if we know the weight (or mass) 
of the body, the velocity in feet per second, and the duration 
of the impact in time. Remember that the mass of any body 
is its weight divided by 32.2; that is, anything that weighs 
just 96.6 pounds, we would say that its mass was 3. 



CENTRIFUGAL FORCE. 

The centrifugal force of a fly wheel of an engine, or any- 
thing of similar shape, is the force exerted outward from 
the center, like the sparks from an emery wheel; the force of 
which can be found (approximately) as follows: We have an 
engine with a fly wheel, the rim of which weighs 60 tons, or 
120000 pounds, is 18 feet in diameter at the center of the rim 
and has a velocity of 80 revolutions per minute. Required, 
the centrifugal force. 

18 feet the diameter=56.5 feet in circumference, which 
multiplied by the number of revolutions 80=4520 feet per 
minute, and divided by 60=75.3 feet per second; 75.3 
squared=5670, and multiplied by the weight in pounds= 
680400000, and dividing this by the radius in feet, multiplied 
by 32.2, is the answer. Thus the radius of the wheel is 9 
feet, which multiplied by 32.2=about 290; then 68040000O di- 
vided by 290=2346206 pounds, the answer. 



220 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CHAPTER VI. 



SCREW CUTTING. 

In nearly all modern lathes threads can be cut of most 
any pitch by simple gearing, by which we mean that there is 
one driver and one driven gear, and usually but one inter- 
mediate gear placed between them; but it makes no differ- 
ence how many there are so long as they connect one with 
another in a direct train. The number of teeth in the inter- 
mediate gears also do not affect it; only the driver, as it is 
called, on the spindle and the driven on the leading screw 
are to be considered. 

If the leading screw has eight threads per inch, and we 
wish to cut eight threads on a piece of work, then, as usually 
made, any two gears with the same numbers of teeth, one on 
the spindle and the other on the leading screw will cut it. If 
the leading screw has 4 threads per inch and we want to cut 8 
threads, it is plain that the leading screw should turn only 
one-half as fast as the work, for the reason that while the 
leading screw makes four revolutions, and consequently the 
carriage travels one inch, the piece of work on the centers 
must make twice as many revolutions, or eight turns, in order 
to have eight threads per inch. 

In simple gearing, then, if the leading screw has four 
threads per inch and we wish to cut four threads per inch on 
a piece of work, any two gears with the same numbers of teeth 
will answer. 

And, if the leading screw has four threads and we want 
to cut five threads per inch, multiplying these numbers by any 
other number will answer, thus: 4X6=24 and 5X6=30; then 
24 and 30 will answer; or, 4X7=28 and 5X7=35; then 28 and 
■35 will cut it, and so on. 

Remember to multiply the number of threads on the lead- 
ing screw and the number of threads you wish to cut by the 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 221 

same number, no matter what the number may be; if you 
have the gears, it will be correct. 

Sometimes we find the pair of gears required thus : lead- 
ing screw has two (2) threads per inch and we wish to cut six 
(6) threads per inch; then 2x10^—21 and 6XlO>4=63. It 
is plain to see that one is just three times as coarse as the 
other, and consequently, one gear must have just three times 
as many teeth as the other. It is also plain that if we are 
to cut threads finer than the leading screw that the largest 
gear should always be placed on the leading screw, and when 
coarser than the leading screw, the largest gear should go 
on the spindle. 

SCREW CUTTING BY COMPOUND GEARING. 

When cutting screws that are of a. very coarse pitch or 
very fine it is usually done by compound gearing, for the 
reason that it is difficult to find a pair of gears in which the 
difference in the numbers of teeth is great enough to answer 
the purpose by simple gearing. 

In compound gearing we generally have four (4) gears, 
arranged in two pairs as follows: Gear on spindle connects 
with one on stud, and this is one pair; by the side of this gear 
on the stud is another gear that meshes with the gear on the 
leading screw, this is the second pair. Always, in com- 
pound gearing, these four gears have to be considered both 
in regard to the numbers of teeth, as well as being placed in 
their proper positions. 

Suppose, for instance, that the leading screw of a lathe 
has four (4) threads per inch and we want to cut a coarse 
thread on a worm, say one thread or one turn in three (3) 
inches; then, while the worm is making just one turn on the 
centers the screw will have to make twelve turns, and con- 
sequently, one is exactly twelve (12) times as coarse as the 
other, or, as we usually say, the compound ratio of the gears 
to cut this worm is 12 to 1. 

Now we can take a pair, say 20 and 60 for one pair, which 
is a ratio of 3 to 1, and dividing the compound ratio 12 by 



222 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

the simple ratio of 3 we have four for the answer, which 
means that the next pair must be 4 to 1, and it makes no dif- 
ference what the numbers of their teeth may be in either case 
so that they are kept in pairs and proportioned 3 to 1 and 4 
to 1. It is plain to see, however, that in both cases the largest 
wheels are the drivers; thus 60 can be placed on the spindle 
and 20 on the stud for one pair, and 72 on the stud by the side 
of the 20 and 18 on the leading screw; or, 72 can be placed 
on the spindle connecting with 18 on the stud and 60 along- 
side of the 18 in mesh, with 20 on the leading screw. Or, 
we can take a pair of gears 2 to 1, say 40 and 80, and dividing 
12 by 2, we have 6 for the ratio of the next pair; then any 
pair 20 and 120 or 30 and 180 will answer. 

Suppose that the leading screw on a lathe has four (4) 
threads per inch and we want to cut a thread of Y% inch 
pitch. In a case of this kind we generally find how many 
threads there will be in a certain number of inches, both on 
the leading screw and the work that is to be threaded. Let 
us take three (3) inches in this example, and we find that there 
are 24 eighths, and the pitch is to be three-eighths; then there 
will be just eight full threads in three inches, while in the 
leading screw of four threads per inch there will be just 
twelve full threads in three inches, then the gears to cut this 
screw will be as 8 is to 12; that is, in simple gearing where 
only two gears are changed, multiply 8 and 12 by 2, 3, 4, 
5, 6, or any other number, sometimes by 2 l / 2i 3 J /2, 4j4, and 
so on, and if you have such gears they will be correct. Or, 
if you compound the gears, take one pair alike, say 60 and 60 
or 50 and 50, and the second pair will be as just described in 
simple gearing (8 to 12); for instance, 24 and 36. 

Or, suppose you want to cut a screw of % inch pitch, 
the leading screw being four threads per inch; then we find 
that in 7 inches there are 56 eighths, and as each thread 
takes up a space of % inches, dividing 56 by 7 we have just 
eight full threads in the seven inches, while in the leading 
screw there will be 7 times 4, or 28 full threads; then our 
ratio will be as 8 is to 28, or 3J4 to 1. In simple gearing we 



THE MACHINIST AND TOOI, MAKER' S INSTRUCTOR. 223 

can take any gear, say 20 for one and S J / 2 times 20 for the 
next and so on, while in compound gearing we can take a pair 
of 2 to 1 and dividing 3 l / 2 by 2, thus: 3.5-^2=1.75, or 1J4 to 
1, for the next pair; thus any pair 2 to 1, call it 30 and 60 for 
one pair, and 40 and 70 for the next. 

Suppose the leading screw has three (3) threads per inch, 
and we want to cut a screw of Y% inch pitch. In three inches 
there are 24 eighths, and dividing 24 eighths by three-eighths 
we have eight full threads in three inches; while in the lead- 
ing screw there would be 9 full threads in three inches, and 
the gear should be as 8 is to 9; then in simple gearing mul- 
tiply these numbers by any other number, say 3, and we have 
24 and 27; or, in compound gearing, these same gears will 
answer for one pair if the next pair are alike. 

If we should want to cut 27 threads to the inch and the 
leading screw has 3 threads per inch, then 27-^3=9 for the 
ratio; then any pair 3 to 1, say 20 and 60 for one pair, and 
dividing the ratio of 9 by 3, we then have two pairs of gears 
3 to 1; while if we should take for one pair 4 to 1, say 22 
and 88, or 20 and 80, then dividing the ratio of 9 by 4, we 
have for the second pair 2J4 to 1; that is, any number in 
which one gear has 2% times as many teeth as the other, for 
instance, 24 and 54. 

Again, suppose our leading screw has four threads per 
inch and we want to cut a coarse thread on a worm of 2^ 
inch pitch; then while the worm makes four revolutions the 
carriage will have to move four times 2% inches or 10y 2 
inches, and, as the leading screw has four threads per inch, it 
will turn four times 10y 2 , or 42 revolutions; the compound 
ratio of the gears will then be 42 revolutions divided by 4 
revolutions, or 10^; we will now take 30 and 90 for the one 
pair, which is 3 to 1, and dividing 10.5 by 3 and we will have 
thus: 10.5-^-3=3.5, or S J / 2 to 1 for the second pair; thus 20 
and 70 or 24 and 84 will answer. 



224 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CHAPTER VII. 



A Method of Calculating the Speed and the Diame- 

* 

ter of Pulleys. 

The countershaft of a lathe, or any other machine, has a 
pulley 16" diameter and should run 80 revolutions per min- 
ute; the main shaft runs at 100 revolutions per minute. What 
should be the size of pulley on the main shaft? 

Multiply the diameter of the driven pulley by the speed 
it should run per minute and divide the product by the speed 
of the main shaft, thus: 

16"=di. of pulley on counter shaft. 
80=revolutions per minute. 

Revolutions of 

main shaft=100)1280(12.8:=diameter of pulley on 
100 main shaft. 

280 
200 

800 
800 



We have an emery wheel that should run 3,600 revolu- 
tions per minute; the pulley on the emery wheel spindle is 
3 inches diameter and the pulley on the countershaft is 14 
inches diameter that is to drive the 3-inch pulley; alongside 
this 14" pulley on the counter shaft is a pulley 6 inches in 
diameter; the main shaft runs at 120 revolutions per minute. 
Required, the diameter of pulley on main shaft. 

Dividing the 14" pulley on the countershaft by the 3" 
pulley on the emery wheel spindle we have as follows: 14-f- 
3=4 2-3, or 4.66, which means that the speed of the counter- 
shaft will be as 4.66 is contained in 3600, which is 772.5; then 
dividing 772.5 by 120, the speed of the main shaft, and we 
have 6.43, which means that the pulley on the main shaft 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 225 

should be 6.43 times as large as the pulley on the counter- 
shaft which it is to drive; then multiplying tne 6-inch pulley 
by 6.43 we have 38.5+, or 38y 2 inches for the size of the pulley 
required. 

Another way: Multiply the speed required by the diam- 
eter of the driven pulley on the emery wheel spindle and di- 
vide by the driver on the countershaft; multiply this by the 
driven pulley on the countershaft and divide by the speed 
of the main shaft, thus: 3600X3=10800, and 10800-5-14=771.4; 
771.4X6=4628.4; then 4628.4-7-120=38.5, the answer. 

The countershaft of a machine has no pulley on it, and 
we want it to run at a speed of 240 revolutions per minute; 
the main shaft has a pulley 18 inches diameter and has a speed 
of 130 revolutions per minute. Required, the diameter of 
pulley on the countershaft. Multiply the diameter of the 
driver by its number of revolutions per minute and divide 
this by the number of revolutions required. Thus: 
130 revolutions of main shaft. 
18 diameter of pulley. 

1040 
130 

240)2340(9.75 inches, the answer. 
2160 

1800 
1680 

1200 
1200 

We have a machine that should run at a speed of 800 
revolutions per minute; the countershaft has no pulleys, but 
we have a pulley 20 inches diameter on the main shaft which 
runs at a speed of 130 revolutions per minute; our machine 
has a pulley 5 inches diameter on the spindle. What size 
pulleys may we select for the countershaft in order to get the 
speed required? 

Dividing the speed required for the machine by the 'speed 



226 THE MACHINIST AND TOOL MAKER' S INSTBUCTOR. 

of the main shaft, thus: 800-^-130=6.15, we find that the 
ratio of speed is as 6.15 to 1. Dividing the diameter of 
pulley on the main shaft by the one on the spindle, thus: 20 
-r-5=4, and we find the ratio of the two pulleys is as 4 to 1. 
Therefore, dividing the ratio of the speed by the ratio of the 
pulleys, thus: 6.15-^4=1.53. Then we may select any pulley, 
say 16 inches for the large one, and dividing 16 bv 1.53= 
10.4 inches, which will be the diameter of the other pulley. That 
is, a belt connecting a 20-inch pulley on the main shaft (130 
per minute) with the 10.4 inch pulley on the countershaft, and 
another belt connecting the 16-inch pulley alongside of the 
10.4" pulley on the countershaft with the 5" pulley on the ma- 
chine will drive it with a speed of 800 revolutions per minute. 

We have a machine that should run at a speed of 240 revo- 
lutions per minute; it has no pulley on the spindle, and we 
wish to use a countershaft that has two pulleys, one of 12 and 
the other 8" diameter; there is also a pulley 14" diameter on 
the main shaft, the speed of which is 140 revolutions per min- 
ute; required, the diameter of the pulley on the machine. 

Dividing the speed of machine by the speed of the main 
shaft, thus: 240-^140=about 1.7 for the ratio of speed required. 

As the speed of the machine should be greater than the 
main line, we will drive from the 14" pulley on the main shaft 
to the 8" pulley on the countershaft. Then dividing 14 by 8 
we have a ratio of 1.75, and, as this is greater than what we 
want, (1.7), then the pulley on the machine should be a little 
larger than the one that is to drive it (the 12") and dividing 
the ratio of speed that the 14" pulley would drive the 8" (1.75) 
by the ratio required (1.7), and the answer 1.03Xl2"=12.3" 
about, for the size required. Or thus, 140X14=1960-^8= 
245X12=2940-^-12.3=239-)-, the speed of the machine per 
minute. 

The simplest manner, then, to find the diameter of one or 
more pulleys when the speed of the main shaft and also the 
machine that you wish to drive is given, is as follows : 

The main shaft revolves at a speed of 80 revolutions per 
minute, and we wish to drive the spindle of the machine with 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Tli 

a velocity of 2300 revolutions per minute; required, the size 
of pulleys. 2300-f-80=about 28.7. Now if we want only two 
(2) pulleys, one of them would be 28.7 times as large as the 
other; it would not make any difference what it was, so that 
the large one was 28.7 times the size of the small pulley. But 
it would be much better to use four pulleys, one on the main 
shaft, two on the countershaft and one on the spindle. In 
such a case we should try not to have them vary in size any 
more than possible, and, therefore, we would take for the first 
pair a ratio of 5 to 1; that is, say one of 6 inches and the other 
30 inches diameter, for the one pair, and dividing 28.7 by the 
ratio of 5, we have for the next pair a ratio of 5.7+; that is, 
say the small pulley is 5 inches diameter, then the large pulley 
will be 5.7 times 5, or about 28.5 inches in diameter. 



CHAPTER VIII. 



Condensed Suggestions for Steei, Workers. 
Crescent Steei, Co., Pittsburg, Pa. 

(by permission.) 

ANNEALING. 

Owing to the fact that the operations of rolling or hammer- 
ing steel make it very hard, it is frequently necessary that the 
steel should be annealed before it can be conveniently cut into 
the required shapes for tools. 

Annealing or softening is accomplished by heating steel 
to a red heat and then cooling it very slowly, to prevent it 
from getting hard again. 

The higher the degree of heat the more will steel be sof- 
tened, until the limit of softness is reached, when the steel is 
melted. 

It does not follow that the higher a piece of steel is heated 
the softer it w T ill be when cooled, no matter how slowly it may 
be cooled; this is proved by the fact that an ingot is always 
harder than a rolled or hammered bar made from it. 



228 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

Therefore, there is nothing gained by heating a piece of 
steel hotter than a good, bright, cherry red; on the contrary, a 
higher heat has several disadvantages; first, if carried too far, 
it may leave the steel actually harder than a good red heat 
would leave it; second, if a scale is raised on the steel, this 
scale will be harsh, granular oxide of iron, and will spoil the 
tools used to cut it. It often occurs that steel is scalded 
in this way, and then because it does not cut well, it is custom- 
ary to heat it again, and hotter still, to overcome the trouble, 
while the fact is, that the more this operation is repeated, the 
harder the steel will work, because of the hard scale and the 
harsh grain underneath; third, a high scaling heat, continued 
for a little time, changes the structure of the steel, destroys 
its crystalline property, makes it brittle, liable to crack in 
hardening and impossible to refine. 

Again, it is common practice to put steel into a hot fur- 
nace at the close of a day's work and leave it there all night. 
This method always gets the steel too hot, always raises a 
scale on it, and worse than either, it leaves it soaking in the 
fire too long, and this is more injurious to steel than any other 
operation to which it can be subjected. 

A good illustration of the destruction of crystalline struc- 
ture by long continued heating may be had by operating on 
chilled cast iron. 

If a chill be heated red hot and removed from the fire as 
soon as it is hot, it will, when cold, retain its peculiar crystal- 
line structure; if now it be heated red hot, and left at a moder- 
ate heat for several hours — in short, if it be treated as steel 
often is, and be left in the furnace over night, it will be found 
when cold, to have a perfect amorphous structure, every trace 
of chill crystals will be gone and the whole piece will be non- 
crystalline gray cast iron. If this is the effect upon coarse 
cast iron, what better is to be expected from fine cast steel? 

A piece of fine tap steel, after having been in a furnace 
over night, will act as follows: 

It will be harsh in the lathe and spoil the cutting tools. 

When hardened it will almost certainly crack; if it does 



THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 229 

net crack it will be a remarkably good steel to begin with. 
When the temper is drawn to the proper color and the tap is 
put into use, the teeth will either crumble off or crush down 
like so much lead. 

Upon breaking the tap, the grain will be coarse and the 
steel brittle. 

To anneal any piece of steel, heat it red hot; heat it uni- 
formly and heat it through, taking care not to let the ends 
and corners get too hot. 

As soon as it is hot, take it out of the fire, the sooner the 
better, and cool it as slowly as possible. A good rule for 
heating is to heat it at so low a red that when the piece is 
cold it will still show the blue gloss of the oxide that was put 
there by the hammer or the rolls. 

Steel annealed in this way will cut very soft; it will harden 
very hard, without cracking, and when tempered it will be 
very strong, nicely refined, and will hold a keen, strong edge. 

HEATING TO FORGE. 

Fully as much trouble and loss are caused by improper 
heating in the forge fire as in the tempering fire, although 
steel may be heated safely very hot for forging if it be done 
properly; but any high degree of heat, no matter how uniform 
it may be, is unsafe for hardening. 

The trouble in the forge fire is usually uneven heat, and 
not too high heat. Suppose the piece to be forged has been 
put into a very hot fire, and forced as quickly as possible to 
a high yellow heat, so that it is almost up to the scintillating 
point. If this be done, in a few minutes the outside will be 
quite soft and in nice condition for forging, while the middle 
parts will be not more than red hot. The highly heated soft 
outside will have very little tenacity; that is to say, this part 
will be so far advanced toward fusion that the particles will 
slide easily over one another, while the less highly heated 
inside parts will be hard, possessed of high tenacity, and the 
particles will not slide so easily over each other. 

The soft outside will yield so much more readily than 



230 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 

the hard inside that the outer particles will be torn asunder, 
while the inside will remain sound, and the piece will be 
pitched out and branded "burned." 

Suppose the case to be reversed, and the inside to be 
much hotter than the outside; that is, that the inside shall be 
in a state of semi-fusion, while the outside is hard and firm. 

If now the piece be forged, the outside will be all sound 
and the whole piece will appear perfectly good until it is 
cropped, and then it is found to be hollow inside, and it is 
pitched out and branded "burned." 

In either case, if the piece had been heated soft all through, 
or if it had been only red hot all through, it would have 
forged perfectly sound and good. 

If it be asked, why then is there ever any necessity for 
smiths to use a low heat in forging, when a uniform high heat 
will do as well: We answer: 

In some cases a high heat is more desirable to save heavy 
labor, but in every case where a fine steel is to be used for cut- 
ting purposes, it must be borne in mind that very heavy 
forging refines the bars as they slowly cool, and if the smith 
heats such refined bars until they are soft, he raises the grain, 
makes them coarse, and he cannot get them fine again unless 
he has a very heavy steam hammer at command, and knows 
how to use it well. 

In following the above hints there is a still greater 
danger to be avoided; that is by letting the steel lie in the fire 
after it is properly heated. When the steel is hot through 
it should be taken from the fire immediately and forged as 
quickly as possible. 

"Soaking" in the fire causes steel to become dry and 
brittle, and does it more injury than any bad practice known 
to the most experienced. 

HEATING. 
Owing to the varying instructions on a great many differ- 
ent labels, we find at times a good deal of misapprehension as 
to the best way to heat steel; in some cases this causes too 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 231 

much work for the smith, and in other instances disasters fol- 
low the act of hardening. 

There are three distinct stages or times of heating: 

First, for forging, 

Second, for hardening, 

Third, for tempering. 

The first requisite for a good heat for forging is a clean 
fire and plenty of fuel, so that jets of hot air will not strike the 
corners of the piece; next, the fire should be regular, and 
give a good uniform heat to the whole part to be forged. It 
should be keen enough to heat the piece as rapidly as may be, 
and allow it to be thoroughly heated through, without being 
so fierce as to overheat the corners. 

Steel should not be left in the fire any longer than is nec- 
essary to heat it clear through, as "soaking " in fire is very 
injurious; and, on the other hand, it is necessary that it 
should be hot through to prevent surface cracks, which are 
caused by the reduced cohesion of the overheated parts, which 
overlie the colder center of an irregularly heated piece. 

By observing these precautions a piece of steel may al- 
ways be heated safely up to even a bright yellow heat when 
there is much forging to be done on it, and at this heat it 
will weld well. 

The best and most economical of welding fluxes is clean, 
crude borax, which should be first thoroughly melted and then 
ground to a fine powder. Borax prepared in this way will not 
froth on the steel, and one-half of the usual quantity will do 
the work as well as the whole quantity unmelted. 

After the steel is properly heated, it should be forgred to 
shape as quickly as possible, and just as the red heat is leaving 
the parts intended for cutting edges these parts should be 
refined by rapid light blows, continued until the red disap- 
pears. 

For the second stage of heating, for hardening, great 
care should be used; first, to protect the cutting edges and 
working parts from heating more rapidly than the body of the 
piece; next, that the whole part to be hardened be heated 



232 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

uniformly through, without any part becoming visibly hotter 
than the other. A uniform heat, as low as will give the re- 
quired hardness, is the best for hardening. 

BEAR IN MIND, 

That for every variation of heat which is great enough 
to be seen there will result a variation in grain, which may 
be seen by breaking the piece; and for every such variation 
in temperature there is a very good chance for a crack to be 
seen; many a costly tool is ruined by inattention to this point, 

The effect of too high heat is to open the grain; to make 
the steel coarse. 

The effect of an irregular heat is to cause irregular 
grain, irregular strains and cracks. 

As soon as the piece is properly heated for hardening, 
it should be promptly and thoroughly quenched in plenty 
of the cooling medium; water, brine, or oil, as the case may 
be. An abundance of the cooling bath, to do the work 
quickly and uniformly all over, is very necessary to good and 
safe work. To harden a large piece safely, a running stream 
should be used. Much uneven hardening is caused by the 
use of too small baths. 

For the third stage of heating, to temper, the first im- 
portant requisite is again uniformity. The next is time. The 
more slowly a piece is brought down to its temper, the better 
and safer is the operation. 

When expensive tools, such as taps, reamers, etc., are to 
be made, it is a wise precaution, and one easily taken, to try 
small pieces of steel at different temperatures, so as to find out 
how low a heat will give the necessary hardness. The low- 
est heat is the best for any steel; the test costs nothing, takes 
very little time, and very often saves considerable loss. 

TEMPER. 
The word temper, as used by the steel maker, indicates 
the amount of carbon in steel; thus, steel of high temper is 
steel containing much carbon; steel of low temper is steel 



THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 233 

containing little carbon; steel of medium temper is steel con- 
taining carbon between these limits, etc., etc. Between the 
highest and the lowest we have some twenty divisions, each 
representing a definite content of carbon. 

As the temper of steel can only be observed in the ingot, 
it is not necessary to the needs of the trade to attempt any 
description of the mode of observation, especially as this is 
purely a matter of education of the eye, only to be obtained 
by years of experience. 

Likewise, the quality of steel cannot be determined from 
the appearance of the fracture of a bar as it comes from the hands 
of the manufacturer. This appearance is determined in the 
main, by the heat at which the bar is finished, and therefore, 
one end of a long bar (and especially of a hammered bar) may 
show a coarse, and the other end, a fine grain, where the 
whole bar will be well suited for the purpose intended. Two 
tools properly heated, forged and hardened (one from each 
end of such a bar) will, if broken, show fractures sirnlar in 
color and grain. 

The act of tempering steel is the act of giving to a piece 
of steel, after it has been shaped, the hardness necessary for 
the work it has to do. This is done by first hardening the 
piece, generally a good deal harder than is necessary, and then 
toughening it by slow heating and gradual softening, until it 
is just right for work. 

A piece of steel properly tempered should always be fin- 
ished finer in grain than the bar from which it is made. If it is 
necessary, in order to make the piece as hard as is required, 
to heat it so hot that after being hardened it will be as coarse, 
or coarser, in grain than the bar, then the steel itself is of too 
low temper for the desired work. In a case of this kind, the 
steel maker should at once be notified of the fact, and could 
immediately correct the trouble by furnishing higher steel. 

Sometimes an effort is made to harden fine steel without 
removing (by grinding or other method) the scale formed in 
rolling, hammering or annealing. The result will generally 
be disappointing, as steel which would harden through such 



234 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 

a coating would be of too high a temper where the scale was 
removed. 

This surface scale is necessarily of irregular thickness and 
density, is oxide of iron — not steel — and, therefore, it will not 
harden, and is to a certain extent a bad conductor of heat. 
It should, therefore, be removed in every case to insure the 
best results. 

If a great degree of hardness is not desired, as in the case 
of taps, and most tools of complicated form, and it is found 
that at a moderate heat the tools are too hard and are liable to 
crack, the smith should first use a lower heat in order to save 
the tools already made, and then notify the steel maker that his 
steel is too high, so as to prevent a recurrence of the trouble. 
In all cases where steel is used in large quantities for the same 
purpose, as in the making of taps, reamers, drills, etc., there 
is very little difficulty about temper, because, after one or two 
trials, the steel maker learns what his customer requires and 
can always furnish it to him. 

In large general works, however, such as a rolling mill 
and nail factory, or large machine works, or large railroad 
shops, both the maker and worker of the steel labor under 
great disadvantages from want of a mutual understanding. 

The steel maker receives his order and fills the sizes, of 
temper best adapted to general work, and the smith usually 
tries to harden all tools at about the same heat. The steel 
maker is right, because he is afraid to make the steel too high 
or too low, for fear it will not suit, and so he gives an average 
adapted to the size of the bar. The smith is right, because he 
is generally the most hurried and crowded man about the es- 
tablishment. He must forge a tap for this man, a cold nail 
•knife for that one, and a lathe tool for another, and so on; 
and each man is in a hurry. Under these circumstances he 
cannot be expected to stop and test every piece of steel he 
uses, and find out exactly at what heat it will harden best and 
refine properly. He needs steel that will all harden properly 
at the same heat, and this he usually gets from the general 
practice among steel makers of making each bar of a certain 
temper, according to its size. 



THE MACHINIST AND TOCXL MAKERS INSTRUCTOR. 235- 

But if it should happen that he were caught with only one 
bar of say, inch and a quarter octagon, and three men should 
come in a hur-y, one for a tap, another for a punch and an- 
other for a chilled roll plug, he would find it very difficult to 
make one bar of steel answer for all of these purposes even if 
it were of the very be#t quality. The chances are that he 
would make one good tool and two bad tools; and when the 
steel maker came around to inquire, he would find one friend 
and two enemies, and the smith puzzled and in doubt. 

There is a perfectly easy and simple way to avoid all of this 
trouble; and that is, to write after each size the purpose for 
which it is wanted, as for instance, lathe tools, taps, dies, hot 
or cold punches, shear knives, etc. This gives very little 
trouble in making the order, and it is the greatest relief to 
the steel maker. It is his delight to get hold of such an order, 
for he knows that when it is filled he will hardly ever hear a. 
complaint. 

Every steel maker worthy of the name knows exactly 
what temper to provide for any tool, or if it is a new case, 
one or two trials are enough to inform him, and as he always 
should have twenty odd tempers on hand, it is just as easy, 
and far more satisfactory to both parties, to have it made right 
as to have it made wrong. 

For these reasons we urge all persons to specify the work 
the steel is to do, then the smith can harden all tools at about 
the same heat, and he will not be annoyed by complaints, or 
hints that he does not do his work well. 



MISCELLANEOUS. 



(From The Pratt & Whitney Co.) 

For comprehensive information regarding the subject of 
standard pipe and pipe threads, as applied to American prac- 
tice, we would refer all who may be interested to the Excerpt 
Minutes of Proceedings of the Institution of Civil Engineers 



236 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 

-of Great Britain, Vol. LXXI, Session 1882-3, Part I, contain- 
ing the papers of the late Robert Briggs, C. E., presented and 
read after his death, on "American Practice in Warming 
Buildings by Steam.'' 

The following extracts from the paper of Mr. Briggs (in- 
cluded more fully in the report of the committee on standard 
pipe and pipe threads, American Society of Mechanical Engi- 
neers, Vol. VIII, transactions,) are here presented, giving 
data upon which the Briggs standard pipe-thread sizes are 
oased. 

The taper employed for the conical tube ends is uniform 
with all makers of tubes or fittings, namely, an inclination of 
1 in 32 to the axis. Custom has also established a particular 
length of screwed end for each different diameter of tube. 
Tubes of the several diameters are kept in stock by manu- 
facturers and merchants, and form the basis of a regular trade 
in the apparatus for warming by steam. A knowledge of all 
these particulars is therefore essential for designing apparatus 
for the purpose. The ruling dimensions in wrought iron 
tube work is the external diameter of certain nominal sizes, 
which are designated roughly according to their internal 
diameter. These nominal sizes were mainly established in 
the English tube trade between 1820 and 1840, and certain 
pitches of screw-thread w r ere adopted for them, the coarseness 
of the pitch carrying roughly with the diameter, but in an 
arbitrary way utterly devoid of regularity. The length of the 
screwed portion on the tube end varies with the external 
diameter of the tube according to an arbitrary rule of 
thumb, whence results, for each size of tube, a certain mini- 
mum of thickness of metal at the outer extremity of the taper- 
ing screwed tube-end. It is the determination of this mini- 
mum thickness of metal, for the tapering screwed end of a 
wrought-iron tube, which constitutes the question of mechani- 
cal interest. 

The thread employed has an angle of 60°; it is slightly 
rounded off, both at the top and at the bottom, so that the 
height or depth of the thread, instead of being exactly equal 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 



237 



to the pitch, is only four-fifths (4-5) of the pitch, or equal to 
0.08 1/n, if n be the number of threads per inch. For the 
length of tube-end throughout which the screw-thread con- 
tinues perfect, the empirical formula used is (0.8D+4.8) Xl/n, 
where D is the actual external diameter of the tube throughout 
its parallel length, and is expressed in inches. Farther back, 
beyond the perfect threads, come two having the same taper 
at the bottom, but imperfect at the top. The remaining im- 
perfect portion of the screw-thread farthest back from the 
extremity of the tube is not essential in any way to this sys- 
tem of joint; and its imperfection is simply incidental to the 
process of cutting the thread at a single operation. 

STANDARD DIMENSIONS OF WROUGHT IRON 
WELDED TUBES. 

(Briggs Standard. n 



DIAMETER OF TUBE. 



") 



SCREWED ENDS. 











NO. OF 
THREADS 


j LENGTH OF 


NOMINAL 


ACTUAL 


ACTUAL 


THICKNESS 


PERFECT 


INSIDE. 


INSIDE. 


OUTSIDE. 


OF METAL. 


THREAD AT 










PER INCH. 


BOTTOM. 


Vs in. 


0.270 in. 


0.405 in. 


0.068 in. 


27 


0.19 in. 


H " 


0.364 " 


0.540 44 


0.088 M 


18 


0.29 M 


fk " 


0.494 M 


0.675 44 


0.091 44 


18 


0.30 44 


{ A " 


623 M 


0.840 4 * 


0.109 44 


14 


039 " 


% " 


0.824 " 


1.050 ' 4 


0113 44 


14 


0.40 M 


1 " 


1.048 " 


1.315 44 


0.134 44 


1V4 

1VA 


0.51 44 


Ui" 


1.380 " 


1.660 44 


0.140 44 


0.54 4< 


VA ' 


1.610 " 


1.900 4< 


0.145 M 


IVA 


0.55 44 


2 " 


2.067 " 


2.375 4I 


0.154 44 


11H 


0.58 44 


2*4 M 


2.468 M 


2.875 " 


0.204 44 


8 


^.89 44 


3 M 


3.067 " 


3.500 '■ 


.217 " 


8 


0.95 4 * 


PA" 


3.548 M 


4.000 44 


.226 ,4 


8 


1.0 ,4 


4 4t 


4.026 l- 


4.500 44 


.237 44 


8 


1.05 44 


4 1 /* " 


4.508 " 


5.000 44 


.246 44 


8 


1.10 " 


5 " 


5.045 4t 


5.563 44 


.259 44 


8 


1.16 * 4 


6 " 


6.065 44 


6.625 44 


.280 44 


8 


1.26 " 


7 M 


7.023 44 


7.625 44 


.301 44 


8 


1.36 M 


8 M 


7.982 4i 


8.625 44 


.322 44 


8 


1.46 4t 


*9 M 


9.000 " 


9.688 M 


.344 44 


8 


1.57 * 4 


10 " 


10.019 4< 


10.750 44 


.366 ' 4 


8 


1.68 ,s 



*By the action of the Manufacturers of Wrought Iron 
Pipe and Boiler Tubes, at a meeting held in New York, May 
9, 1889, a change in size of actual outside diameter of 9 



238 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



inch pipe was adopted, making the latter 9.625 instead of 
9.688 inches, as given in the table of Briggs standard pipe 
■diameters. 

The sizes of Twist Drills to be used in boring holes to be 
reamed with Pipe Reamer, and threaded with Pipe Taps are 
as follows : 



SIZE OF TAP. 


DIAM. OF DRILL 


SIZE OF TAP. 


DIAM. of DRILL. 


y% inch. 

i/ <« 


21/64 inch. 

29/64 " 
19/32 " 
23/32 " 
15/16 " 


1 inch. 

IX " 

2 " 
2/ 2 " 

3 


1 T \ inch. 

115 « 

if! " 

2A " 

Ol 1 " 



The angle for turning or grinding pipe taps and reamers 
is 1° 47'=(angle from center of axis.) 

THE SELLERS SYSTEM. 

Recommended by the Franklin Institute of Philadelphia, 
has been adopted by the United States Government, the Mas- 
ter Mechanics and Master Car Builders Associations, Locomo- 
tive Works, Machine Bolt Makers and by many manufacturing 
establishments throughout the country. The thread has an 
angle of 60 degrees, with flat top and bottom equal to one- 
eighth of the pitch. The advantages of this form of thread 
over the sharp V are that, in the tap, the edges of the thread 
are less liable to accidental injury, and will wear and retain 
their size and form longer, and, in the bolt, the flat top and 
bottom give increased strength and an improved appearance, 
while the greater facility with which practical uniformity and 
consequent interchangeability is now attained by it's use, as 
compared with tne Witworth form, will commend it to the at- 
tention of every user of taps and dies, wherever its application 
may be possible. 

The accompanying table gives the standard diameter and 
number of threads per inch for all usual sizes, from one-quar- 
ter inch to six inches, inclusive. 



THE MACHINIST AND TOOL. MAKER'S INSTRUCTOR. 



239 



SEINERS OR U. S. STANDARD. 

Diameter X A H A % A # ^ ^ 

Ko. Threads per inch 20 18 16 14 13 12 11 10 9 

Diameter 1 1% 1% 1# 1% IjK 1% 1# 2 

No. Threads per inch 877665^55 4^ 

Diameter 2% 2% 2)i 2% 2/ s $U 2% 3 

No. Threads per inch , 4^ 4^ 4 4 4 i Z% Z% 

Diameter 3^ 3X 3^ 3^ 3>£ 3^ 3^ 4 

No. Threads per inch 3>£ %y 2 3X 3^ 3^ 3 3 3 

Diameter 4/ s ±% 4}i 4 l A 4# 4# 4^ 5 

No. Threads per inch 2^ 2j£ 2^ 2# 2>£ 2^ 2# 2% 

Diameter h% b% 5^ b l / z Q'/& o% oji 6 

No. Threads per inch 2/ 2 2% 2j/ s 2^ 2y % 2yi 2% 2% 

A 6 4, A 50 > # 40 > A 36 > A 32 and A 28 are al so according to 
the Sellers system „ 

Screws and bolts 11-16, 13-16, and 15-16-inch diameter 
are usually made, having 11, 10 and 9 threads per inch, respect- 
ively, but under the Sellers formula, strictly followed, they 
should be 10, 9 and 8, respectively. 



DIMENSIONS OF PRATT & WHITNEY CO. REAMERS 

FOR MORSE STANDARD TAPER TWIST 

DRILIv SOCKET. 



No 


Diam. 
small 
end. 


Diam. 

large 

end. 


Gauge 

Diam. 

large 

end. 


Gauge 
length. 


Length 

of 
Flutes. 


Total 
length 


Taper 
per 
foot. 


Center 
Angle or 
Angle to 

grind. 




in. 


in. 


in. 


in. 


in. 


in. 


in. 




1 


0.365 


0.525 


0.475 


t% 


3 


5tf 


0.605=1°27 / — 


2 


0.573 


0.749 


0.699 


i% 


s% 


S* 


0.600=1°26'— 


3 


0.779 


0.982 


0.936 


3^ 


4 


ny 2 


0.605=1°27 / — 


4 


1.026 


1.283 


1.231 


4 


5 


$u 


0.615=1°28 / 


5 


1.486 


1.796 


1.746 


5 


6 


10 


0.625=1°29 / + 


6 


2.117 


2.566 


2.500 


7X 


sy 2 


ny 2 


0.634=1°31 / — 



240 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



USEFUL INFORMATION. 
STEAM. 
(By permission of the Geo. F. Blake Mfg. Co., New York.) 

A cubic inch of water evaporated under ordinary atmos- 
pheric pressure is converted into 1 cubic foot of steam (ap- 
proximately). 

The specific gravity of steam (at atmospheric pressure) 
is .411 that of air at 34° Fahrenheit, and .0006 that of water 
at the same temperature. 

27.222 cubic feet of steam weigh one (1) pound; 13,817 
cubic feet of air weight one (1) pound. 

Locomotives average a consumption of 3000 gallons of 
water per 100 miles run. 

The best designed boilers, well set, with good draft, and 
skillful firing, will evaporate from 7 to 10 pounds of water 
per pound of first class coal. 

In calculating horse-power of tubular or flue boilers, con- 
sider 15 square feet of heating surface equivalent to one nomi- 
nal horse-power. 

On one square foot of grate can be burned on an average 
from 10 to 12 pounds of hard coal, or 18 to 20 pounds soft coal, 
per hour, with natural draft. With forced draft nearly double 
these amounts can be burned. 

Steam engines, in economy, vary from 14 to 60 pounds of 
feed water and from \y 2 to 7 pounds of coal per hour per in- 
dicated H.-P. See table following for duty of high grade 
engines. 

Condensing engines require from 20 to 30 gallons of 
water at an average low temperature, to condense the steam 
represented by every gallon of water evaporated in the boilers 
supplying engines — approximately for most engines, we say, 
from 1 to iy 2 gallons condensing water per minute per in- 
dicated horse-power. 

Surface condensers should have about 2 square feet of 
tube (cooling) surface per horse-power for a compound steam 



THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 241 

engine. Ordinary engines will require more surface accord- 
ing to their economy in the use of steam. It is absolutely 
necessary to place air pumps below condensers to get satis- 
factory results. 

RATIO OF VACUUM TO TEMPERATURE 
(Fahrenheit) OF FEED WATER. 

INCHES VACUUM. 

00 212° 

11 190° 

18 170° 

22^ 150° 

*25 135° 

27 tf 112° 

28# 92° 

29 72° 

29J£ 52° 

♦Usually considered the standard point of efficiency— Condenser 
and Air Pump being well proportioned. 

WEIGHT AND COMPARATIVE FUEL VALUE 
OF WOOD. 

1 cord air-dried hickory or hard maple weighs ^ about 
4,500 pounds, and is equal to about 2,000 pounds of coal. 

1 cord air-dried white oak weighs about 3,850 pounds, 
and is equal to about 1,715 pounds of coal. 

1 cord air-dried beech, red oak and black oak, weighs 
about 3,250 pounds, and is equal to about 1,450 pounds of 
coal. 

1 cord air-dried poplar (white wood), chestnut and elm, 
weighs about 2,350 pounds, and is equal to about 1,050 pounds 
of coal. 

1 cord air-dried average pine, weighs about 2,000 pounds 
and is equal to about 925 pounds of coal. 

From the above it is safe to assume that 2% pounds of 
dry wood is equal to 1 pound average quality of soft coal, and 
that the full value of the same weight of different woods is 
very nearly the same; that is, a pound of hickory is worth no 



242 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



more for fuel than a pound of pine, assuming both to be dry. It 
is important that the wood be dry, as each 10 per cent, of 
water or moisture in wood will detract about 12 per cent, from 
its value as fuel. 



DUTY OF STEAM ENGINES. 

A well known engineer of high authority gives the fol- 
lowing comparative figures, showing the economy of high 
grade steam engines in actual practice: 



Cost per I. 
H. P. per 
hour sup- 
posing 
coal at $6 
per ton. 



TYP3 OF 
KNGINK. 



Tempera- 
ture of 
feed water 



Pouuds of 

water 
evaporated 
per lb. of 
Cumber- 
land Coal. 



Pounds of 

steam per 

I. H. P. 

used per 

hour. 



Pounds of 
Cumber- 
land Coal 
used per I. 
H. P. per 
hour. 



Non- 
Condensing, 

Condensing, 

Compound 
Jacketed, 

Triple 
Expansion 
Jacketed, 



210° 
100° 

100° 
100° 



10.5 
9.4 

9.4 
9.4 



29. 
20. 

17. 
13.6 



2.75 
2.12 

1.81 
1.44 



$0.0073 
0.0056 

0.0045 
0.0036 



The effect of a good condenser and air pump should be 
to make available about 10 pounds more mean effective pres- 
sure, with the same terminal pressure; or to give the same 
mean effective pressure with a correspondingly less terminal 
pressure. When the load on the engine requires 20 pounds 
M. E. P., the condenser does half the work; at 30 pounds, 
one-third of the work; at 40 pounds, one-fourth, and so on. 
It is safe to assume that practically the condenser will save from 
one-fourth to one-third of the fuel, and it can be applied to 
any engine, cut-off, or throttling, where a sufficient supply 
of water is available. 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 243 



USEFUL INFORMATION.— WATER. 
(Geo. F. Blake Mfg. Co. 

Doubling the diameter of a pipe increases its capacity- 
four times. Friction of liquids in pipes increases as the 
square of the velocity. 

The mean pressure of the atmosphere is usually estimated 
at 14.7 pounds per square inch, so that with a perfect vacuum 
it will sustain a column of mercury 29.9 inches, or a column 
of water 33.9 feet high at sea level. 

To find the pressure in pounds per square inch of a col- 
umn of water, multiply the height of the column in feet by 
.434. Approximately, we say that every foot elevation is equal 
to y 2 pound pressure per square inch; this allows for ordinary 
friction. 

To find the diameter of a pump cylinder to move a given 
quantity of water per minute (100 feet of piston being the 
standard of speed) divide the number of gallons by 4, then ex- 
tract the square root, and the product will be the diameter 
in inches of the pump cylinder. 

To find quantity of water elevated in one minute running 
at 100 feet of piston speed per minute, square the diameter 
of the water cylinder in inches and multiply by 4. Example : 
Capacity of a 5-inch cylinder is desired. The square of the 
diameter (5 inches) is 25, which multiplied by 4 gives 100, the 
number of gallons per minute (approximately.) 

To find the horse-power necessary to elevate water to a 
given height, multiply the weight of the water elevated per 
minute in pounds by the height in feet, and divide the pro- 
duct by 33,000 (an allowance should be added for water fric- 
tion and a further allowance for loss in steam cylinder, say 
from 20 to 30 per cent.) 

The area of the steam piston, multiplied by the steam pres- 
sure, gives the total amount of pressure that can be exerted. 
The area of the water piston multiplied by the pressure of 
water per square inch, gives the resistance. A margin must 



244 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 

be made between the power and the resistance to move the 
pistons at the required speed — say from 20 to 40 per cent., ac- 
cording to speed and other conditions. 

To find the capacity of a cylinder in gallons, multiply- 
ing the area in inches by the length of stroke in inches will 
give the total number of cubic inches; divide this amount by 
231 (which is the cubical contents of a U. S. gallon in inches) 
and the product is the capacity in gallons. 



WEIGHT AND CAPACITY OF DIFFERENT STANDARD 
GALLONS OF WATER. 

Cubic inches Weight of a Gal- Gallons in a 
in a Gallon. Ion in pounds. Cubic Foot. 

Imperial or English 277.274 10.00 6.232102 

United States 231 8.33111 7.480519 

Weight of a cubic foot of water, English Standard, 62.321 
lbs. Avoirdupois. 

Weight of Crude Petroleum, 6)4 lbs. per U. S. gallon. 

Weight of Refined " 6*4 lbs. per U. S. gallon. 

42 gallons to the barrel. 

A "miner's inch" of water is approximately equal to a 
supply of 12 U. S. gallons per minute. 



THE MACHINIST AND TOOL MAKERS' INSTRUCTOR. 



245 



TABLE OF SQUARES AND HEXAGONS, EXACT SIZE, 
ACROSS FLATS AND CORNERS. 




STAND- 


ACROSS 
CORNERS. 


STAND- 


LARGEST 


LARGEST 


STAND- 


ACROSS 


ARD 
SQUARE. 


ARD 

ROUND. 


INSCRIBED 
SQUARE. 


INSCRIBED 
HEXAGUN. 


ARD 
HEXAGON. 


CORNERS. 


1/8 


.1767 


1/8 


.0S83 


.108 


1/8 


.1443 


5/32 


.2209 


5/32 


.1104 


.1352 


5/32 


.1803 


3/16 


.2651 


3/16 


.1325 


.1623 


3/16 


.2165 


7/32 


.3093 


7/32 


.1546 


.1893 


7/32 


.2526 


1/4 


.3535 


1/4 


.1767 


.2165 


1/4 


.2887 


5/16 


.4419 


5/16 


.2209 


.2706 


5/16 


.3608 


3/8 


.5303 


3/8 


.2651 


.3247 


3/8 


.4330 


7/16 


.6187 


7/16 


.3093 


.3788 


7/16 


.5052 


1/2 


.7071 


1/2 


.3535 


.4330 


1/2 


.5773 


9/16 


.7955 


9/16 


.3977 


.4871 


9/16 


.6495 


5/8 


.8839 


5/8 


.4419 


.5412 


5/8 


.7217 


11/16 


.9723 


11/16 


.4861 


.5953 


11/16 


.7938 


3/4 


1.0606 


3/4 


.5303 


.6495 


3/4 


.8660 


13/16 


1.1490 


13/16 


.5745 


.7036 


13/16 


.9382 


7/8 


1.2374 


7/8 


.6187 


.7577 


7/8 


1.0104 


15/16 


1.3258 


15/16 


.6629 


.8118 


15/16 


1.0825 


1 


1.4142 


1 


.7071 


.8660 


1 


1.1547 


1A 


1.5026 


ItV 


.7513 


.9201 


iA 


1.2269 


l* 


1.5910 


H 


.7954 


.9742 


H 


1.2990 


1A 


1.6794 


1A 


.8396 


1.0283 


iA 


1.3712 



246 



THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 



TABLE OF SQUARES AND HEXAGONS, EXACT SIZE, 
ACROSS FLATS AND CORNERS. 



STAND- 


ACROSS 
CORNERS. 


STAND- 


LARGEST 


LARGEST 


STAND- 


A fIDACQ 


ARD 

SQUARE. 


ARD 
ROUND. 


INSCRIBED 
SQUARE. 


INSCRIBED 
HEXAGON. 


ARD 
HEXAGON 


AUKUoo 

CORNERS. 


H 


1.7677 


It 


.8838 


1.0825 


n 


1.443 


1A 


1.8561 


lr 5 s 


.9280 


1.1366 


1A 


1.5155 


if 


1.9445 


If 


.9722 


1.1907 


if 


1.5877 


1A 


2.0329 


i* 


1.016 


1.2448 


1A 


1.6599 


n 


2.1213 


if 


1.060 


1.2990 


ii 


1.7320 


1 9 
X J<5 


2.2097 


i& 


1.104 


1.3531 


1 9 
1 T¥ 


1.8042 


1 5 


2.2981 


if 


1.149 


1.4072 


1 5 


1.8764 


113 


2.3865 


m 


1.193 


1.4613 


111 


1.9486 


if 


2.4748 


it 


1.237 


1.5155 


If 


2.0207 


HI 


2.5632 


m 


1.281 


1.5696 


1H 


2.0929 


1 7 


2.6516 


n 


1.325 


1.6237 


if 


2.1651 


115 

J-Te 


2.7400 


m 


1.370 


1.6778 


itf 


2.2372 


2 


2.8284 


2 


1.414 


1.7320 


2 


2.3094 


2i 


3.0052 


2i 


1.502 


1.8402 


2f 


2.4537 


2i 


3.1819 


2i 


1.590 


1.9485 


n 


2.5980 


2f 


3.3587 


2f 


1.679 


2.0567 


2f 


2.7424 


» 


3.5355 


2i 


1.767 


2.1650 


2| 


2.8867 


2f 


3.7123 


2f 


1.856 


2.2732 


2 5 - 


3.0311 


2| 


3.8890 


2| 


1.944 


2.3815 


H 


3.1754 


21 


4.0658 


2J 


2.033 


2.4897 


2f 


3.3197 


3 


4.2426 


3 


2.121 


2.5980 


3 


3.4641 


3J 


4.5961 


H 


2.298 


2.8145 


H 


3.7528 


3^ 


4.9497 


3i 


2.474 


3.0310 


H 


4.0414 


3| 


5.3032 


81 


2.651 


3.2475 


3-i 


4.3301 


4 


5.6568 


4 


2.828 


3.4640 


4 


4.6188 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 247 

TAPERS PER INCH AND CORRESPONDING ANGLES, 
ALSO THE ANGLES TO TURN OR GRIND. 



TAPER PER 
INCH. 


INCLUDED 
ANGLE. 


ANGLE TO 

TURN OR 

GRIND. 


TAPER PER 
INCH. 


INCLUDED 
ANGLE. 


ANGLE TO 

TURN OR 

GRIND. 




1-64" 


0° 


54' 


0° 


27'- 


33-64" 


28° 


54' 


14° 


27'+ 


1-32" 




1° 


48' 


0° 


54'- 


17-32 


29° 


46' 


14° 


53'— 




3-64 


2° 


42' 


1° 


21'— 


35-64 


30° 


36' 


15° 


18'— 


1-16 




3° 


34' 


1° 


47'+ 


9-16 


31° 


24' 


15° 


42'+- 




5-64 


4° 


28' 


2° 


14' 


37-64 


32° 


14' 


16° 


7'+ 




3-32" 


5° 


22' 


2° 


41' 


19-82" 


33° 


4' 


16° 


32'+ 


7-64" 




6° 


16' 


3° 


8'- 


39-64 


33° 


54' 


16° 


57'- 




1-8 


7° 


10' 


3° 


35'- 


5-8 


34° 


42' 


17° 


21'+ 


9-64 




8° 


2' 


4° 


1'+ 


41-64 


35° 


32' 


17° 


46'— 




5-32 


8° 


56' 


4° 


28' 


21-32 


36° 


20' 


18° 


10'- 


11-64" 




9° 


50' 


4° 


55'— 


43-64" 


37° 


8' 


18° 


34'+ 




3-16" 


10° 


42' 


5° 


21'+ 


11-16 


37° 


56' 


18° 


58'+ 


13-64 




11° 


36' 


5° 


48'- 


45-64 


38° 


44' 


19° 


22'+ 




7-32 


12° 


28' 


6° 


14'+ 


23-32 


39° 


32' 


19° 


46' 


15-64 




13° 


22' 


6° 


41' 


47-64 


40° 


20' 


20° 


1C- 




1-4 " 


14° 


16' 


7° 


8'— 


3-4 " 


41° 


6' 


20° 


33'+ 


ft-04" 




15° 


8' 


7° 


34'- 


49-64 


41° 


54' 


20° 


57'- 




9-32 


16° 


0' 


8° 


0'+ 


25-32 


42° 


40' 


21° 


20'+ 


19-64 




16° 


54' 


8° 


27'- 


51-64 


43° 


26' 


21° 


43'4 




5-16 


17° 


46' 


8° 


53'— 


13-16 


44° 


14' 


22° 


7'— 


21-64" 




18° 


38' 


QO 


19' f 


53-64" 


44° 


58' 


22° 


29'+ 




11-32" 


19° 


30' 


9° 


45'+ 


27-32 


45° 


44' 


22° 


52'+ 


23-64 




20° 


22' 


10° 


11'+ 


55-64 


46 a 


30' 


23° 


15'+ 




3-8 


21° 


14' 


10° 


37'+ 


7-8 


47° 


16' 


23° 


38' 


25-64 




22° 


6' 


11° 


3' 


57-64 


48° 


0' 


24° 


0'+ 




13-32" 


22° 


58' 


11° 


29'— 


29-32" 


48° 


46' 


24° 


23'- 


27-64" 




23° 


48' 


11° 


54'+ 


59-64 


49° 


30' 


24° 


45'- 




7-16 


24° 


40' 


12° 


20'+ 


15-16 


50° 


14' 


25° 


7'- 


29-64 




25° 


32' 


12° 


46'— 


61-64 


50° 


58' 


25° 


29'- 




15-32 


26° 


22' 


13° 


11'+ 


31-32 


51° 


42' 


25° 


51'- 


31-64" 




27° 


14' 


13° 


37'— 


63-64" 


52° 


24' 


26° 


12'+ 




1-2" 


28° 


4' 


14° 


2'+ 


1 


53° 


8' 


26° 


34' 



248 THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 

TAPERS PER FOOT AND CORRESPONDING ANGLES, 
ALSO THE ANGLES TO TURN OR GRIND. 



TAPER PER 
FOOT. 


INCLUDED 
ANGLE. 


ANGLE TO 

TURN OR 

GRIND. 


TAPER PER 
FOOT. 


INCLUDED 
ANGLE. 


ANGLE TO 
TURN OR 
GRIND. 




1-8 " 


0° 


34' 


0° 


17'+ 


1 " 


4° 


46' 


2° 


23'+ 


9-64" 




0° 


40' 


0° 


20'+ 


1 1-16 


5° 


4' 


2° 


32'+ 




5-32 


0° 


44' 


0° 


22'+ 


1 1-8 


5° 


22' 


2° 


41' 


11-64 




0° 


50' 


0° 


25'- 


1 3-16 


5° 


40' 


2° 


50'— 




3-16 


0° 


54' 


0° 


27'- 


1 1-4 


5° 


58' 


2° 


59'— 




13-64" 


0° 


58' 


0° 


29'+ 


1 5-16" 


6° 


16' 


3° 


8'- 


7-32" 




1° 


2' 


0° 


31'+ 


1 3-8 


6° 


34' 


3° 


17'- 




15-64 


1° 


8' 


0° 


34'— 


1 7-16 


6° 


52' 


3° 


26'- 


1-4 




1° 


12' 


0° 


36'— 


1 1-2 


7° 


10' 


3° 


35'— 




17-64 


1° 


16' 


0° 


38'+ 


1 9-16 


7° 


26' 


3° 


43'+ 


9-32" 




1° 


20' 


0° 


40' + 


1 5-8 " 


7° 


44' 


3° 
4° 

4° 
4° 
4° 


52'+ 
1'+ 
10'+ 
19'+ 
28' 




19-64" 


1° 


26' 


0° 


43'- 


1 11-16 


8° 


2' 


5-16 




1° 


30' 


0° 


45'— 


1 3-4 


8° 


20' 




21-64 


1° 


34' 


0° 


47' 


1 13-16 


8° 


38' 


11-32 




1° 


38' 


0° 


49'+ 


1 7-8 


8° 


56' 




23-64" 


1° 


42' 


0° 


51'+ 


1 15-16" 


9° 


14' 


4° 


37'— 


3-8 " 




1° 


48' 


0° 


54'— 


2 


9° 


32' 


4° 


46'- 




25-64 


1° 


52' 


0° 


56'- 


2 1-8 


10° 


8' 


5° 


4'- 


13-32 




1° 


56' 


0° 


58'+ 


2 1-4 


10° 


42' 


5° 


21'+ 




27-64 


2° 


0' 


1° 


0'+ 


2 3-8 


11° 


18' 


5° 


39'+ 


7-16" 




2° 


4' 


1° 


2'f 


2 1-2 " 


11° 


54' 


5° 


57'- 




29-64" 


2° 


10' 


1° 


5'- 


2 5-8 


12° 


30' 


6° 


15'-- 


15-32 




2° 


14' 


1° 


7'+ 


2 3-4 


13° 


4' 


6° 


32'+ 




31-64 


2° 


18' 


1° 


9'+ 


2 7-8 


13° 


40' 


6° 


50'— 


1-2 




2° 


24' 


1° 


12'— 


3 


14° 


16' 


7° 


8'— 




17-32" 


2° 


32' 


1° 


16'+ 


3 1-8 " 


14° 


50' 


7° 


25'+ 


9-16" 




2° 


42' 


1° 


21'- 


3 1-4 


15° 


26' 


7° 


43'- 




19-32 


2° 


50' 


1° 


25' 


3 3-8 


16° 


0' 


8° 


0'+ 


5-8 




2° 


58' 


1° 


29'+ 


3 1-2 


16° 


36' 


8° 


18'- 




21-32 


3° 


8' 


1° 


34'- 


3 5-8 


17° 


10' 


8° 


35'+ 


11-16" 




3° 


16' 


1° 


38'+ 


3 3-4 " 


17° 


46' 


8° 


53'- 




23-32" 


3° 


26' 


1° 


43'- 


3 7-8 


18° 


20' 


9° 


10'+ 


3-4 




3° 


34' 


1° 


47' + 


4 


18° 


56' 


9° 


28'- 




25-32 


3° 


41' 


1° 


52'— 


4 1-8 


19° 


30' 


9° 


45'+ 


13-16 




3° 


52' 


1° 


56'+ 


4 1-4 


20° 


4' 


10° 


2'+ 




27-32" 


4° 


2' 


2° 


1'— 


4 1-2" 


21° 


14' 


10° 


37'+ 


7-8" 




4° 


10' 


2° 


5'+ 


4 3-4 


22° 


24' 


11° 


12'- 




29-32 


4° 


20' 


2° 


10'- 


5 


23° 


32' 


11° 


46' 


15-16 


31-32 


4° 
4° 


28' 

38' 


2° 

2° 


14'+ 
19'— 













THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



249 



TAP DRILLS FOR SHARP V AND U. S. STANDARD 
TIJREADS. 



The sizes of Twist Drills given in these Tables are correct 
for general purposes. 







NO. OF 
THREADS 
PES INCH 


DIAMETER OF 


SIZE OF 


DIAMETER OF 

TAP AT 

BOTTOM OF 

U. S. STAND. 

THREAD. 


SIZE OF 


DIAMETER 
OF TAP. 


TAP AT 
BOTTOM OF 
V THREAD. 


DRILL FOR 

V 
THREAD. 


DRILL FOR 

U.S. STAND. 

THREAD. 




f 


16 


.142" 


11-64" ) 






1-4" 


{ 

f 


18 
20 
16 


.154 

.163 
.173 


3-16 [ 
3-16 ) 
13-64 


.185" 


No. 11. 


9-32 


{ 


18 
20 


.185 
.194 


7-32 
7-32 






5-16 


I 


16 
18 


.204 
.216 


15-64 1 
1-4 J 


.240 


1-4" 


11-32 


\ 


16 
18 
14 


.235 

.247 
.251 


17-64 
9-32 
19-64 ) 






3-8 


( 


16 
18 
14 


.267 
.279 
.282 


19-64 y 

5-16 J 
21-64 


.294 


19-64 


13-32 


X 


16 
18 


.298 
.310 


21-64 
11-32 






7-16 


\ 


14 
16 


.313 
.329 


23-64 X 
23-64 J 


.344 


23-64 


15-32 


{ 

( 


14 
16 
12 


.344 

.360 
.356 


25-64 
25-64 
13-32 ) 






1-2 


f 


13 
14 
12 


.366 
.376 

.387 


27-64 Y 
27-64 J 
7-16 


.399 


13-32 


17-32 


1 


13 
14 


.397 
.407 


7-16 
29-64 






9-16 


I 


12 
14 


.418 
.438 


15-32 \ 
31-64 j 


.454 


15-32 


19-32 


{ 


12 

14 


.449 
.469 


1-2 
33-64 






5-8 


1 


10 
11 


.452 
.468 


1-2 ) 

33-64 Y 


.507 


33-64 




12 


.481 


17-32 ) 








! 


10 


.483 


17-32 






21-32 


11 


.499 


35-64 








12 


.512 


9-16 







250 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



TAP DRILLS FOR SHARP V AND U. S. STANDARD 
THREADS. 







[CONTINUED.] 






DIAMETER 
OF TAP. 


NO. OF 

THREADS 
PER INCH. 


DIAMETER OF 

TAP AT 

BOTTOM OF 

V THREAD. 


SIZE OF 

DRILL FOR 

V 

THREAD. 


DIAMETER OF 

TAP AT 

BOTTOM OF 

U. S. STAND. 

THREAD. 


SIZE OP 

DRILL FOR 

U.S. STAND 

THREAD. 


11-16" { 


11 
12 


.53( " 

.543 


37-64" 
19-32 






H { 


10 

11 

' 12 


.577 
.593 
.606 


5-8 ) 
41-64 y 
21-82 j 


.62 


5-8" 


25-32 -j 


10 
11 
12 


.608 
.624 
.637 


21-32 
43-64 
11-16 






13-16 { 


9 
10 


.620 
.639 


43-64 
11-16 






7-8 { 


9 
10 


.683 
.702 


47-64 1 
3-4 ; 


.731 


47-64 


29-32 { 


9 
10 


.714 
.733 


49-64 
25-32 






15-16 j 


8 
9 


.720 
.745 


25-32 
51-64 






1 


8 


.783 


27-32 


.837 


27-32 


If { 


7 
8 


.878 
.908 


61-64 "I 
31-32 J 


.940 


61-64 


1 5 J 


7 
8 


.909 
.939 


63-64 
1 






u 


7 


1.003 


1 5 


1.065 


1& 


1 9 


7 


1.034 


1*¥ 






11 


6 


1.086 


m 


1.158 


m 


113 
1^2 


6 


1.117 


igf 






H 


6 


1.211 


m 


1.283 


hi 


117 
-I 35 


6 


1.242 


m 






« { 


5 
5| 


1.279 
1.311 


Hi I 
iii J 


1.389 


m 


11 


5 


1.404 


ill 


1.490 


H 


« i 


5 


1.491 
1.529 


HI 1 


1.615 


if 


2 


ih 


1.616 


Iff 


1.712 


Iff 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



251 



TAP DRILLS FOR MACHINE SCREWS. 

The sizes of Drills given in this Table correspond to the 
Brown & Sharpe Twist Drill and Steel Wire Gauge. 









DIAMETER OF 




NUMBER 


THREADS PER 


DIAMETER 


TAP AT BOTTOM 


SIZE OF DRILL 


OF TAP. 


INCH. 


OF TAP. 


OF THREAD. 


FOR TAPPING. 


4 


32 


.110 


.056 


No. 45 


4 


36 


.110 


.062 


44 44 


4 


40 


.110 


.067 


44 44 


6 


30 


.137 


.080 


No. 37 


6 


32 


.137 


.083 


* 35 


6 


36 


.137 


.089 


" 35 


6 


40 


.137 


.094 


44 as 


8 


24 


.163 


.091 


No. 30 


8 


30 


.163 


.106 


44 29 


8 


32 


.163 


.109 


" 29 


8 


36 


.163 


.115 


44 29 


10 


20 


.189 


.103 


No. 29 


10 


22 


.189 


.111 


44 28 


10 


24 


.189 


.117 


" 24 


10 


30 


.189 


.132 


44 22 


10 


32 


.189 


.135 


" 20 


12 


20 


.216 


.130 


No. 20 


12 


22 


.216 


.138 


44 19 


12 


24 


.216 


.144 


44 15 


14 


16 


.242 


.134 


No. 19 


14 


18 


.242 


.146 


44 16 


14 


20 


.242 


.156 


44 13 


14 


22 


.222 


.164 


44 10 


14 


24 


.242 


.170 


44 5 


16 


16 


.268 


.160 


No. 11 


16 


18 


.268 


.172 


44 6 


16 


20 


.268 


.182 


•• 3 


16 


22 


.268 


.190 


44 2 


18 


16 


.295 


.187 


No. 2 


18 


18 


.295 


.199 


44 1 


18 


20 


.295 


.209 


11 15-64 


20 


16 


.321 


.213 


No. 1-4 


20 


18 


.321 


.225 


44 1-4 


20 


20 


.321 


.235 


44 17-64 


22 


16 


.347 


.239 


No. 17-64 


22 


18 


.347 


.251 


44 9-32 


24 


14 


.374 


.250 


No. 19-64 


24 


16 


.374 


.266 


44 19-64 


24 


18 


.374 


.278 


44 5-16 


26 


16 


.400 


.292 


No. 21-64 l 



252 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



TABLE OF DECIMAL EQUIVALENTS OF STUBS STEEI 
WIRE GAUGE. 

The Crescent Drill Rods made by Miller, Metcalf and Parkin, Pitts 
T)urgh, Pa., also correspond to these sizes. 



LETTER. 


SIZE OF 
LETTER 
IN DECI- 
MALS. 


NO. OP 
WIRE 
GAUGE. 


SIZE OF 
NUMBER 
IN DECI- 
MALS. 


NO. OP 
WIRE 
GAUGE. 


SIZE OF 
NUMBER 
IN DECI- 
MALS. 


NO. OF 

wirh: 

GAUGE. 


SIZE OF 
NUMBER 
IN DECI- 
MALS. 


z 


.413 


1 


.227 


28 


.139 


55 


.050 


Y 


.404 


2 


.219 


29 


.134 


56 


.045 


X 


.397 


3 


.212 


30 


.127 


57 


.042 


W 


.386 


4 


.207 


31 


.120 


58 


.041 


V 


.377 


5 


.204 


32 


.115 


59 


.040 


TJ 


.368 


6 


.201 


33 


.112 


60 


.039 


T 


.358 


7 


.199 


34 


.110 


61 


.038 


S 


.348 


8 


.197 


35 


.108 


62 


.037 


B 


.339 


9 


.194 


36 


.106 


63 


.036 


Q, 


.332 


10 


.191 


37 


.103 


64 


.035 


P 


.323 


11 


.188 


38 


.101 


65 


.033 


O 


.316 


12 


.185 


39 


.(199 


66 


.032 


N 


.302 


13 


.182 


40 


.097 


67 


.031 


M 


.295 


14 


.180 


41 


.095 


68 


.030 


L 


.290 


15 


.178 


42 


.092 


69 


.029 


K 


.281 


16 


.175 


43 


.088 


70 


.027 


J 


.277 


17 


.172 


44 


.085 


71 


.026 


I 


.272 


18 


.168 


45 


.081 


72 


.024 


H 


.266 


19 


.164 


46 


.079 


73 


.023 


G 


.261 


20 


.161 


47 


.077 


74 


.022 


F 


.257 


21 


.157 


48 


.075 


75 


.020 


E 


.250 


22 


.155 


49 


.072 


76 


.018 


D 


.246 


23 


.153 


50 


.069 


77 


.016 


C 


.242 


24 


.151 


51 


.066 


78 


.015 


B 


.238 


25 


.148 


52 


.063 


79 


.014 


A 


.234 


26 


.146 


53 


.058 


80 


.013 






27 


.143 


54 


.055 







THE BROWN AND SHARPE TWIST DRILX AND STEEL, WIRE 
GAUGE. 

Size of Numbers. 





SIZE OF 




SIZE OF 




SIZE OF 




SIZE OF 


NO. 


NUMBER IN 


NO. 


NUMBER IN 


NO. 


NUMBER IN 


NO. 


NUMBER IN 




DECIMALS. 




DECIMALS. 




DECIMALS. 




DECIMALS. 


1 


.2280 


16 


.1770 


31 


.1200 


46 


.0810 


2 


.2210 


17 


.1730 


32 


.1160 


47 


.0785 


3 


.2130 


18 


.1695 


33 


.1130 


48 


.0760 


4 


.2090 


19 


•1660 


31 


.1110 


49 


.0730 


5 


.2055 


20 


.1610 


35 


.1100 


50 


.0700 


6 


.2040 


21 


.1590 


36 


.1065 


51 


.0670 


7 


.2010 


22 


.1570 


37 


.1040 


52 


.0635 


8 


.1990 


23 


.1540 


38 


.1015 


53 


.0595 


9 


.1960 


24 


.1520 


39 


.0995 


54 


.0550 


10 


.1935 


25 


.1495 


40 


.0980 


55 


.0520 


11 


.1910 


26 


.1470 


41 


.0960 


56 


.0465 


12 


.1890 


27 


.1440 


42 


.0935 


57 


.0430 


13 


.1850 


28 


.1405 


43 


.0890 


58 


.0420 


14 


.1820 


29 


.1360 


44 


.0860 


59 


.0410 


15 


.1800 


30 


.1285 


45 


.0820 


60 


.0400 





THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 


253 


DIFFERENT STANDARDS FOR 


WIRE GAUGE IN USE 


IN THE 






UNITED 


STATES. 








Dimensions of Sizes in Decimal Parts of an Inch. 








(Brown & Sharpe Mfg. Co.) 






03 K . 

fH 05 W 

w fe 2 
53 &■ 2 


tf O fc Ph 

«og3 


fcOgJH 


1 *** y, 6 

oq 55 2 . 

•« £ C (25 


gss» 


ce ^ ^ 


m. * 




000000 












.4687 


«~ *, & 


00000 








.45 




.4375 


•g<»5 


0000 


.46 


.454 


.3938 


.4 




.4062 


je a '-& 


000 


.40964 


.425 


.3625 


.36 




.375 




00 


.3648 


.38 


.3310 


.33 




.3437 





.32486 


.34 


.3065 


.305 




.3125 


MCC-tf 


1 


.2893 


.3 


.2830 


.285 


.227 


.2812 


IJjy 2 


2 


.25763 


.284 


.2625 


.265 


.219 


.2656 


3*3 


3 


.22942 


.259 


.2437 


.2JL5 


.212 


.25 


4 


.20431 


.238 


.2253 


.225 


.207 


.2343 


% tn^ 


5 


.18194 


.22 


.2070 


.205 


.204 


.2187 


~*o 


6 


.16202 


.203 


.1920 


.19 


.201 


.2031 


s« £ 


7 


.14428 


.18 


.1770 


.175 


.199 


.1875 


a Er 


8 


.12849 


.165 


.1620 


.16 


.197 


.1718 


$■-$ 


9 


.11443 


.148 


.1483 


.145 


.194 


.1562 


£- 


10 


.10189 


,134 


.1350 


.13 


.191 


.1406 


M £ ii . 


11 


.090742 


.12 


.1205 


.1175 


.188 


.125 




12 


.080808 


.109 


.1055 


.105 


.185 


.1093 


g pj 


13 


.071961 


.095 


.0915 


.0925 


.182 


.0937 


8 fc i 


14 


.064084 


.083 


.0800 


.08 


.18) 


.0781 


29 g.-3 


lo 


.057068 


.072 


.0720 


.07 


.178 


.0703 


1 S ~ -5 


36 


.05082 


.065 


.0625 


.061 


.175 


.0625 


8«SPa 


17 


.045257 


.058 


.0540 


.0525 


.172 


.0562 




18 


.040303 


.049 


.0475 


.045 


.168 


.05 


s*n 


19 


.03589 


.042 


.0410 


.04 


.164 


.0437 


* « u 


20 
21 


.031961 
.f 28462 


.035 
.032 


.0348 
.03175 


.035 
.031 


.161 
.157 


.0375 
.0343 


22 


.025347 


.028 


.0286 


.028 


.155 


.0312 


23 


.022571 


.025 


.0258 


.025 


.153 


.0281 


bjO ^ Ik 


24 


.0201 


.022 


.0230 


.0225 


.151 


.025 


p'd'd u 


25 


.0179 


.02 


.0204 


.02 


.148 


.0218 


/11 ctf .„ « 


26 


.01594 


.018 


.0181 


.018 


.146 


.0187 


27 


.014195 


.016 


.0173 


.017 


.143 


.0171 


2 ..2>» 


28 


.012641 


.014 


.0162 


.016 


.139 


.0156 




29 


.011257 


.013 


.0150 


.015 


.134 


.0140 


30 


.010025 


.012 


.0140 


.014 


.127 


•Q25 


0^"^ 


31 


.008928 


.01 


.0132 


.013 


.120 


.0109 


£°s£ 


32 


.00795 


.009 


.0128 


.012 


.115 


.0101 


- Sc-g 


33 


.00708 


.008 


.0118 


.011 


.112 


.0i.93 


3.3 « 3 


34 


.006304 


.007 


.0104 


.01 


.110 


.0086 


35 


.005614 


.005 


.0095 


.0095 


.108 


.0078 


36 


.005 


.004 


.0090 


.009 


.106 


.0070 




37 
38 
39 
40 


.004453 
.003965 
.003531 
.003144 






.0085 
.008 
.0075 
.007 


.103 
.101 
.099 
.097 


.0066 
.0062 


^J V* ? '■" 






^Sfc 






*8? 








So 3 


1 







254 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CIRCUMFERENCES AND AREAS OF CIRCLES. 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER. 


ENCE. 


AREA. 


1/16 


.1963 


.00307 


4 


12.5664 


12.5664 


% 


.3927 


.01227 


4^ 


12.9591 


13.3641 


H 


.7854 


.04908 


m 


13.3518 


14.1863 


% 


1.1781 


.11045 


Ws 


13.7445 


15.0330 


Mi 


1.5708 


.19635 


V/2 


14.1372 


15.9043 


Yb 


1.9635 


.30679 


4% 


14.5299 


16.8002 


% 


2.3562 


.44179 


4% 


14.9226 


17.7206 


% 


2.7489 


.60132 


m 


15.3153 


18.6655 


1 


3.1416 


.7854 


5 


15.7080 


19.6350 


m 


3.5343 


.9940 


Ws 


16.1007 


20.6290 


1M 


3.9270 


1 .2272 


fyi 


16.4934 


21.6476 


Wa 


4.3197 


1.4849 


r o% 


16.8861 


22.6907 


m 


4.7124 


1.7671 


% 


17.2788 


23.7583 


m 


5.1051 


2.0739 


&/s 


17.6715 


24.8505 


i% 


5.4978 


2.4053 


*% 


18.0642 


25.9673 


Ws 


5.8905 


2.7612 


5% 


18.4569 


27.1086 


. 2 


6.2832 


3.1416 


6 


18.8496 


28.2744 


2% 


6.6759 


3.5466 


V/s 


19.2423 


29.4648 


2M 


7.0686 


3.9761 


614 


19.6350 


30.6797 


Ws 


7.4613 


4.4301 


6% 


20.0277 


31.9191 


m 


7.8540 


4.9087 


6^ 


20.4204 


33.1831 


2% 


8.2467 


5.4119 


&/s 


20.8131 


34.4717 


2% 


8.6394 


5.9396 


6% 


21.2058 


35.7848 


2% 


9.0321 


6.4918 


6% 


21.5985 


37.1224 


3 


9.4248 


7.0686 


7 


21.9912 


38.4846 


Ws 


9.8175 


7.6699 


m 


22.3839 


39.8713 


3M 


10.2102 


8.2958 


7M 


22.7766 


41.2826 


m 


10.6023 


8.9462 


Ws 


23.1693 


42.7184 


zy* 


10.9956 


9.6211 


m 


23.5620 


44.1787 


3% 


11.3883 


10.3206 


Ws 


23.9547 


45.6636 


m 


11.7810 


11.0447 


m 


24.3174 


47.1731 


Wa 


12.1737 


11.7933 


7% 


24.7401 


48.7071 



THE MACHINIST AND TOOL. MAKER'S INSTRUCTOR. 



255 



CIRCUMFERENCES AND AREAS OF CIRCLES 
[continued.] 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER. 


ENCE. 


AREA. 


8 


25.1328 


50.2656 


12 


37.6992 


113.097 


m 


25.5255 


51.8487 


im 


38.0919 


115.466 


*u 


25.9182 


53.4563 


im 


38.4846 


117.859 


8% 


26.3109 


55.0884 


12% 


38.8773 


120.276 


m 


26.7036 


56.7451 


12^ 


39.270 


122.719 


8% 


27.0963 


58.4264 


12% 


39.6627 


125.185 


m 


27.4890 


60.1322 


12% 


40.0554 


127.676 


s% 


27 8817 


61.8625 


12J/8 


40.4481 


130.192 


9 


28.2744 


63.6174 


13 


40.8408 


132.733 


9% 


28.6671 


65.3968 ! 


13% 


41.2335 


135.297 


9% 


29.0598 


67.2008 ! 


13M 


41.6262 


137.887 


9% 


29.4525 


69.0293 ] 


13% 


42.0189 


140.501 


9% 


29.8452 


70.8823 1 


13% 


42.4116 


143.139 


9% 


30.2379 


72.7599 ! 


13% 


42.8043 


145.802 


9% 


30.6306 


74.6621 1 


13% 


43.1970 


148.490 


9% 


31.0233 


76.5888 


13% 


43.5897 


151.202 


10 


31.4160 


78.5400 


14 


43.9824 


153.938 


mi 


31.8987 


80.5188 


14% 


44.3751 


156.698 


iom 


32.2014 


82.5161 


14M 


44.7678 


159.485 


10% 


32.5941 


84.5409 


14% 


45.1605 


162.296 


io% 


32.9868 


86.5903 


im 


45.5532 


165.130 


10% 


33.3795 


88.6643 


14% 


45.9459 


167.990 


10% 


33.7722 


90.7628 


14% 


46.3386 


170.874 


10% 


34.1649 


92.8858 


14% 


46.7313 


173.782 


11 


34.5576 


95.0334 


15 


47.1240 


176.715 


11% 


34 9503 


97.2055 


15% 


47.5167 


179.673 


11M 


35 3430 


99.4022 


1534 


47.9094 


182.655 


n% 


35.7357 


101.6234 


15% 


48.3021 


185.661 


n% 


36.1284 


103.8691 


15% 


48.6948 


188.692 


n% 


36.5211 


106.1394 


15% 


49.0875 


191.748 


n% 


36.9138 


108.4343 


15% 


49.4802 


194.828 


n% 


37.3065 


110.7537 


15% 


49.8729 


197.933 



256 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CIRCUMFERENCES AND AREAS OF CIRCLES 

[continued.] 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER. 


ENCE. 


AREA. 


16 


50.2656 


201.062 


20 


62.8320 


314.160 


16% 


50.6583 


204.216 


20^ 


63.2247 


318.099 


16% 


51.0510 


207.395 


20M 


63.6174 


322.063 


16% 


51.4437 


210.598 


20% 


64.0101 


326.051 


16% 


51.8364 


213.825 


2oy 2 


64.4028 


330.064 


16% 


52.2291 


217.077 


20% 


64.7955 


334.102 


16% 


52.6218 


220.354 


20% 


65.1882 


338.164 


16% 


53.0145 


223.655 


20% 


65.5809 


342.249 


17 


53.4072 


226.981 


21 


65.9736 


346.361 


17% 


53.7999 


230.331 


21% 


66.3663 


350.497 


17% 


54.1926 


233.706 


2114 


66.7590 


354.657 


17% 


54.5853 


237.105 


21% 


67.1517 


358.842 


17% 


54.9780 


240.529 


213^ 


67.5444 


363.051 


17% 


55.3707 


243.977 


21% 


67.9371 


367.285 


17% 


55.7634 


247.450 


21% 


68.3298 


371.543 


17% 


56.1561 


250.948 


21% 


68.7225 


375.826 


18 


56.5487 


254.469 


22 


69.1152 


380.134 


18% 


56.9415 


258.0 J 6 


22% 


69.5079 


384.466 


18% 


57.3342 


261.587 


22M 


69.9006 


388.822 


18% 


57.7269 


265.183 


22% 


70.2933 


393.203 


18% 


58.1196 


268.803 


22%£ 


70.6860 


397.609 


18% 


58.5123 


272.448 


22% 


71.0787 


402.038 


18% 


58.9050 


276.117 


22% 


71.4714 


406.494 


18% 


59.2977 


279.811 


22% 


71.8641 


410.973 


19 


59.6904 


283.529 


23 


72.2568 


415.477 


19% 


60.0831 


287.272 


23% 


72.6495 


420.004 


19% 


60.4758 


291.040 


2&A 


73.0422 


424.558 


19% 


60.8685 


294.832 


23% 


73.4349 


429.135 


19% 


61.2612 


298.648 


23^ 


73.8276 


433.737 


19% 


61.6539 


302.489 


23% 


74.2203 


438.364 


19% 


62.0466 


306.355 


23% 


74.6130 


443.015 


19% 


62.4393 


310.245 


23% 


75.0056 


447.690 



THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. 



257 



CIRCUMFERENCES AND AREAS OF CIRCLES 
[continued.] 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 




AREA. 


DIAMETER. 




AREA. 




ENCE. 






ENCE. 




24 


75.3984 


452.39 


28 


87.9648 


615.754 


24% 


75.7911 


457.115 


28% 


88.3575 


621.264 


24% 


76.1838 


461.864 


28U 


88.7502 


626.798 


24% 


76.5765 


466.638 


28% 


89.1429 


632.357 


24% 


76.9692 


471.436 


28% 


89.5356 


637.941 


24% 


77.3619 


476.259 


28% 


89.9283 


643.549 


24% 


77.7546 


481.107 


28% 


90.3210 


649.182 


24% 


78.1473 


485.979 


28% 


90.7137 


654.840 


25 


78.540 


490.875 


29 


91.1064 


660.521 


25% 


78.9327 


495.796 


29% 


91.4991 


666.228 


25% 


79.3254 


500.742 


29M 


91.8918 


671.959 


25% 


79.7181 


505.712 


29% 


92.2845 


677.714 


25% 


80.1108 


510.706 


29% 


92.6772 


683.494 


25% 


80.5035 


515.726 


29% 


93.0699 


689.299 


25% 


80.8962 


520.769 


29% 


93.4626 


695.128 


25% 


81.2889 


525.838 


29% 


93.8553 


700.982 


26 


81.6816 


530.930 


30 


94.2480 


706.860 


26% 


82.0743 


536.048 


30% 


94.6407 


712.763 


26% 


82.4670 


541.190 


30M 


95.0334 


718.6^0 


26% 


82.8597 


546.356 


3<% 


95.4261 


724.642 


26% 


83.2524 


551.547 


30% 


95.8188 


730.618 


26% 


83.6451 


556.763 


30% 


96.2115 


736.619 


26% 


84.0378 


562.003 


30% 


96.6042 


742.645 


26% 


84.4305 


567.267 


30% 


96.9969 


74S.695 


27 


84.8232 


572.557 


31 


97.3896 


754.769 


27% 


85.2159 


577.870 


31% 


97.7823 


760.869 


27% 


85.6086 


583.209 


mi 


98.1750 


766.992 


27% 


86.0013 


588.571 


31% 


98.5677 


773.140 


27% 


86.3940 


593.959 


31% 


98.9604 


779.313 


27% 


86.7867 


599.371 


31% 


99.3531 


785.510 


27% 


87.1794 


604.807 


31% 


99.7458 


791.732 


27% 


87.5721 


610 268 


31% 


100.1385 


797.979 



258 THE MACHINIST AND TOOL KAKEE's INSTRUCTOR. 


CIRCUMFERENCES AND AREAS OF CIRCLES. 






[continued.] 








CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER. 


ENCE. 


AREA. 


32 


100.5312 


804.250 


36 


113.098 


1017.878 


32% 


100.9239 


810.545 


36% 


113.490 


1024.960 


32% 


101.3166 


816.865 


36^ 


113.883 


1032.065 


32% 


101.7093 


823.210 


36% 


114.276 


1039.195 


32% 


102.1020 


829.579 


36^ 


114.668 


1046.349 


32% 


102.4947 


835.972 


36% 


115.061 


1053.528 


32% 


102.8874 


842.391 


36% 


115.454 


1060.732 


32% 


103.2801 


848.833 


36% 


115.846 


1067.960 


33 


103.6728 


855.301 


37 


116.239 


1075.213 


33% 


104.0655 


861.792 


37% 


116.632 


1082.490 


33% 


104.4592 


868.309 


37M 


117.025 


1089.792 


33% 


104.8509 


874.850 


37% 


117.417 


1097.118 


33% 


105.2436 


881.415 


373^ 


117.810 


1104.469 


33% 


105.6363 


888.005 


37% 


118.203 


1111.844 


33% 


106.0290 


894.620 


37% 


118.595 


1119.244 


33% 


106.4217 


901.259 


37% 


118.988 


1126.669 


34 


106.814 


907.922 


38 


119.381 


1134.118 


34% 


107.207 


914.611 


38% 


119.773 


1141.591 


34% 


107.600 


921.323 


38M 


120.166 


1149.089 


34% 


107.992 


928.061 


38% 


120.559 


1156.612 


34% 


108.385 


934.822 


38% 


120.952 


1164.159 


34% 


108.778 


941.609 


38% 


121.344 


1171.731 


34% 


109.171 


948.420 


38% 


121.737 


1179.327 


34% 


109.563 


955.255 


38% 


122.130 


1186.948 


35 


109.956 


962.115 


39 


122.522 


1194.593 


35% 


110.349 


969. 


39% 


122.915 


1202.263 


35% 


110.741 


975.909 


39M 


123.308 


1209.958 


35% 


111.134 


982.842 


39% 


123.700 


1217.677 


35% 


111.527 


989.800 


39% 


124.093 


1225.420 


35% 


111.919 


996.783 


39% 


124.486 


1233.188 


35% 


1133.312 


1003.790 


39% 


124.879 


1240.981 


35% 


1110.705 


1110.822 


39% 


125.271 


1248.798 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



259 



CIRCUMFERENCES AND AREAS OF CIRCLES 
[continued.] 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER- 


ENCE. 


AREA, 


40 


125.664 


1256.64 


44 


138.230 


1520.5 


40% 


126.057 


1264.51 


44% 


138.623 


1529.2 


40% 


126.449 


1272.40 


44^ 


139.016 


1537.9 


40% 


126.842 


1280.31 


44% 


139.408 


1546.6 


40% 


127.235 


1288.25 


uy 2 


139.801 


1555.3 


40% 


127.627 


1296.22 


44% 


140.194 


1564.1 


40% 


128.020 


1304.21 


44% 


140.587 


1572.8 


40% 


128.413 


1312.22 


44% 


140.979 


1681.6 


41 


128.806 


1320.26 


45 


141.372 


1590.4 


41% 


129.198 


1328.32 


45% 


141.765 


1599.3 


41% 


129.591 


1336.41 


45^ 


142.157 


1608.2 


41% 


129.984 


1344.52 


45% 


142.550 


1617.1 


41% 


130.376 


1352.66 


45^ 


142.943 


1625.9 


41% 


130.769 


1360.82 


45% 


143.335 


1634.9 


41% 


131.162 


1369.00 


45% 


143.728 


1644.0 


41% 


131.554 


1377.21 


45% 


144.121 


1652.9 


42 


131.947 


1385.5 


46 


144.514 


1661.9 


42% 


132.340 


1393.7 


46% 


144.906 


1670.9 


42% 


132.733 


1401.9 


46M 


145.299 


1680.0 


42% 


133.125 


1410.3 


46% 


145.692 


1689.1 


42% 


133.518 


1418.6 


46y 2 


146.084 


1698.2 


42% 


133.911 


1426.9 


46% 


146.477 


1707.4 


42% 


134.303 


3435.4 


46% 


146.870 


1716.5 


42% 


134.696 


1443.8 


46% 


147.262 


1725.7 


43 


135.089 


1452.2 


47 


147.655 


1734.9 


43% 


135.481 


1460.7 


47% 


148.048 


1744.2 


43% 


135.874 


1469.1 


47^ 


148.441 


1753.4 


43% 


136.267 


1477.6 


47% 


148.833 


1762.7 


43% 


136.660 


1486.2 


47}£ 


149.226 


1772.1 


43% 


137.052 


1494.7 


47% 


149.619 


1781.4 


43% 


137.445 


1503.3 


41'% 


150.011 


1790.7 


43% 


137.838 


1511.9 


47% 


150.404 


1800.1 



260 



THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 



CIRCUMFERENCES AND AREAS OF CIRCLES. 

[continued.] 





CIRCUMFER- 






CIRCUMFER- 




DIAMETER. 


ENCE. 


AREA. 


DIAMETER. 


ENCE. 


AREA. 


48 


150.796 


1809.6 


49 


153.938 


1885.7 


48% 


151.189 


1819.0 


49^ 


154.331 


1895.4 


48% 


151.582 


1828.5 


491/ 4 


154.723 


1905.0 


48% 


151.975 


1837.9 


49^ 


155.116 


1914.7 


48% 


152.367 


1847.5 


49^ 


155.509 


1924.4 


48% 


152.760 


1856.9 


49^ 


155.902 


1934.2 


48% 


153.153 


1866.5 


49% 


156.294 


1944.0 


48% 


153.545 


1876.1 


49% 


156.687 


1953.7 








50 


157.08 


1963.5 



THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 261 



INDEX. 



CHAPTER I. 

ARITHMETIC 5 to 28 

Addition of whole numbers and signs of 5 

Multiplication of whole numbers and signs of 5 

Division of whole numbers and signs of 6 

DECIMALS— 

Addition of 6 

Subtraction of ? 

Multiplication of 7 

Division of 8 

FRACTIONS— 

Addition of 9 

Subtraction of 10 

Multiplication of 12 

Division of 13 

INVOLUTION, or— 

Squares, cubes, etc 14 

EVOLUTION, or— 

Square and cube root 14 to 18 

MENSURATION— 

Angles 18 to 22 

Solids ! 22 to 28 

TRIGONOMETRY 28 

Tangents, Sines, Cosines, and secants, denned. 29 to 33 

Angles of taper work, How to find 34-36 

GEOMETRICAL FIGURES— 

Hexagon, to find the long and short diameters of 37-38 

Squares, to find the long and short diameters of 39 

Octagons, to find the long and short diameters of 39 

Decagon and Dodecagon, to find the long and short diame- 
ters of 40 

SCREW THREADS— 

V threads, to find the depths of 41-42 

U. S. standard, to find the depths of 42-43 

Whitworth, or English standard 44 

CRANKS— 

When two are to be keyed on the same shaft to locate the 
correct angle 44 to 48 

BALLS— 

To find the groove measurements for 48-50 

BELTS— 

To find the length of a cross belt accurately 50 

To find the length of an open belt accurately 52-54 



262 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 

JIGS AND TEMPLETS— 

To lay out and finish accurately 54 to 60, 62 to 65 

Pulley, to find the size of from a broken section of the rim. 61 

Machines, to set at right angles to the main shaft 62 

Natural sines, tangents, cosines and secants, tables of.65 to 110 

CHAPTER II. 

GEARING— 

Classification of 110 

Diametral pitch Hi 

Sizing- of blanks 112-113 

Table of Tooth Dimensions 114 

Distances between centers 115 

Circular Pitch, with Dimensions 115 

Generating Circles 116 

Pin Gearing 117-118 

Single Curve Gears 119 

Double Curve Gears 121-122 

Double Curve Gears, Tables 123 

Rack and Pinion 124-125 

Involute, the perfect 127-128 

Epicycloid, the perfect 129 to 132 

BEVEL WHEELS— 

Mitre gears, how to draw 132-123 

Ang-ular gears, how to draw 134-135 

Bevel gears, shapes of the teeth 135, 137 

Bevel gears, how to find the angles of for turning and cut- 
ting 137-143 

Bevel gears, how to cut the teeth 143, 144 

Bevel gears, how to find the correct angle for swiveling 
the Table to cut the teeth 144 to 147 

spiral gears- 
how to find the diameter of Blanks when cut with stand- 
ard cutters 147 

How to find the thickness of cutter when the Blank is of 

standard size 150 

Right hand and left hand spirals 151 

To find the angle for setting the Table when cutting 151 

WORM GEARING— 

To find the pitch and thickness of thread of a worm 152 

To find the Dimensions of Worm W^heels 152 to 156 

Tooth Dimensions, Table of 154 

How to find the width of thread at top and bottom 156 

How to find the angle for roughing out the teeth of Worm 

Wheels i57 

Attachment for cutting Worm Wheels complete 157-159 

How to cut the teeth of Worm Wheels. 156-157 



THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 263 

CHAPTER III. 

THE MILLING MACHINE 160 

Advantages of small mills over large ones 163 

Relief or clearance of mills 165 

Width of land at top of teeth 165 

Grooving mills should be hollowing 165 

Emery Wheels for grinding 165 

Oil, its use in milling 166 

Lubricant for milling 167 

Work moving with or against the cutters 167 

Speed and feed of cutters 168-169 

Castings and forgings to be pickled 169 

Index Table 170 

Indexing, explanation of 171 

Cutting spirals with Universal Milling Machines 172 to 176 

Selection of gears for cutting spirals 178-179 

To find the correct angle for setting the machine 180 

The best form of spiral mills for fast cutting 181 

Angles, shapes and dimensions of cutters for general pur- 
poses 181 to 185 

Shank mills and special chuck for 185-186 

Special chuck for milling bolt heads 187-188 

Straight line index drilling 189-191 

CHAPTER IV. 
THE UNIVERSAL GRINDING MACHINE— 

Position of the Table for grinding Tapers 198-200 

Position of machine while grinding two Tapers 201 

The manner of holding thin saws and cutters while grind- 
ing 202 

Position in grinding Angular or Taper Cutters 202 

Grinding Twin or Straddle Mills 202 

A list of Emery Wheels for outside grinding 202 

Wheels for internal grinding 203 

CHAPTER V. 
MECHANICS— 

Questions upon the principles of the Lever 204, 209 

Safety Valves 206-208 

The Inclined Plane 209 

The Screw 210 

Wheel and Axle 211 

Wheel Gearing 211-212 

The Screw and Gear 213 

The Hydraulic Press 214 

Brake Horse-Power of an Engine 214 



264 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 



The Horse-Power of an Engine 215 

Pressure on the Guides of an Engine 216 

To find the diameter of a shaft for transmitting power 217 

Impact or Collision of Bodies 218 

Centrifugal Force 219 

CHAPTER VI. 

Screw Cutting by Simple and Compound Gearing 220 to 224 

CHAPTER VII. 
A method of calculating the speed and diameter of Pul- 
leys 224-227 

CHAPTER VIII. 
THE TREATMENT OF STEEL- 

Annealing ; 227 

Heating to Forge 229 

Heating 230 

Temper 232 

MISCELLANEOUS— 

The Manufacture of Pipe and Pipe Threads 235 

Table of standard dimensions of Wrought Iron Tubes 237 

The size of Twist Drills to be used in boring holes to be 

reamed and threaded with Pipe Taps 238 

The Sellers System or the U. S. Standard shape of threads. 238 
Diameters of Taps and number of threads from ^4" to 6", 

inclusive 239 

Dimensions of the Pratt & Whitney Co.'s Reamers for 

Morse Standard Taper Twist Drill Socket 239 

Useful Information— Steam 240 to 243 

Useful Information — Water 243 to 245 

Table of dimensions across flats and corners of Squares 

and Hexagons 245-24G 

Tapers per Inch and corresponding Angles, Table of 

Tapers per Foot and corresponding Angles, Table of c*--248 

Tap Drills, &" to 2" 249-250 

Tap Drills for Machine Screws 251 

Table of Decimal Equivalents of Stubs and the Crescent 

Steel Co.'s Drill Rods .252 

Size of numbers of the Brown & Sharpe Twist Drill and 

Steel Wire Gauge 252 

Different standards for Wire Gauge in the United States.. 253 
Circumferences and Areas of Circles 254 to 260 



■ 



N 



589#' 



